Abby Hatcher: Mr. and Mrs. Hatcher often fill this role to Abby on some occasions. Her mom is used most often.
In the Adventures in Odyssey video "A Shadow of a Doubt", John Avery Whittaker's trial is presided over by one of these. He listens to all testimony presented and is always willing to admit new evidence even after the jury reaches its verdict—when Dylan shows him definitive proof of Whittaker's innocence just as the initial guilty verdict, based on faulty testimonies given by prior witnesses and a damning closing statement full of Blatant Lies by a pitiful excuse for a defense attorney, is being read, he instructs the jury to consider the new evidence as they reconsider their verdict.
Principal Brown is often petty and frivolous, but in "The Apology" he refuses to punish Gumball or Darwin unless he's given solid evidence that they're causing trouble, and he threatens to fire Miss Simian if she keeps trying to frame them, in spite of his personal feelings for her.
While the Doughnut Sheriff is often reckless and the embodiment of Police Are Useless, has his moments of being this, such as in "The Car"; when the Robinsons demand he arrest the Wattersons for destroying their car, he refuses to do so without a warrant, gives the Robinsons multiple warnings to calm down, and ultimately subdues and arrests them once they smash the Wattersons' own car and attack him in a fit of rage.
The Earth King, he even had an Evil Chancellor, Long Feng, who tried to discredit Aang and co. (among other unpleasantness). He listens to Aang and co. even though they fight their way into his throne room, and is eminently reasonable and rational when deciding their case. He eventually agrees to help plan an invasion of the Fire Nation and arrest Long Feng. It's averted in the Promise comics: the Earth King refuses to have an audience with Zuko after the latter withdraws his support of the Harmony Restoration Movement, instead choosing to go to war with the Fire Nation over the colonies.
Katara has become this, the prime example being that unlike the other White Lotus members, who would stop Korra from going to Republic City, she gives her her blessing to go there, remembering how she went on her journey in the original series. That, and she decides to let her son Tenzin train Korra in Airbending.
Lin Beifong may look like a hardhearted lady who continually sticks to the law, but once you get to know her, she's a good ally and will listen well if you have a crime to tell her about.
Fire Lord Izumi firmly establishes herself as one. She readily agrees to lend resources to defend and retaliate against Kuvira, but she is under no illusions about the Fire Nation's reputation and the political fallout of being party to a preemptive strike.
Except for the short part at the beginning where he mistook Ben for a thief, Magister Labrid is fairly reasonable as far as Plumbers go.
Reinrassig III (Reiny) is in charge of his race by the end of their invasion and seeking peace instead of trying to take every species with theirs since they had been Secretly Dying makes him a much more appealing Highbreed Supreme for the reconstituted Highbreed than their previous one.
Codename: Kids Next Door: Numbuh 362 is the Supreme Leader of the KND, and even though she often has to play parent to "eleventy-billion operatives with short attention spans", the safety and wellbeing of her fellow operatives is always her top priority.
Dan Vs.. Normally, the universe hates Dan, and Dan reciprocates, but sometimes the universe throws Dan a bone in the form of this trope.
In "Dan Vs. Traffic", Dan is having a Potty Emergency while stuck in traffic, and breaks several laws in his haste to get home. A cop pulls him over as he's driving on the sidewalk. Dan explains how badly he has to use the bathroom, and the cop simply lets him leave.
In "Dan Vs. Baseball", Dan and Chris go to ludicrous lengths to kidnap the Commissioner of Baseball. However, when Dan actually stops to explain his grievances against baseball (namely that it broke his car's mirror, and it pre-empted his favorite show), the Commissioner immediately gives Dan the money to fix the mirror and promises to make sure the show never gets pre-empted again. (Turns out the Commissioner is also a fan of that show.) So the kidnapping was completely unnecessary.
Jeremiah Burger in "Dan Vs. Burgerphile" understands that customer service is supposed to be about the customer, and chastises Jeff for forgetting the first rule of their trade: the customer is always right. Earlier in the same episode, when Jeff tries to get the police to arrest Dan, they recognize that he's exercising his right to protest, so instead of using force they just ask him politely to leave and then go away once they find a more important issue to deal with.
Daria's lawyer mother Helen is more or less this, at least compared to the other adults. She's very competent at work (if quite workaholic), is more emotionally stable than her borderline Manchild husband Jake (though not without her share of problems) and attempts to be a good mom to both Daria and Quinn.
Family Guy: Mr. Weed, Peter's deceased former boss. From what we saw, he was a pretty fair employer in spite of his comical perversions off duty. Despite Peter's stupidity causing him problems note Telling an inappropriate joke to a female coworker, sleeping on the job, injuring a baseball player hired to win a softball game , he still gave Peter plenty of chances to fix his mistakes, while making perfectly clear he will fire him if he doesn't clean up his act (in at least one instance he does stick to that threat, but of course Status Quo Is God). In his last appearance, he gave Peter a big promotion all because he was treated to a good dinner... just before he choked to death on a dinner roll.
The head of the monastery from Jonny Quest: The Real Adventures episode "Expedition to Khumbu". When the villains try to frame the boys for stealing an artifact from the monastery, he is ready to give them the benefit of doubt, despite the evidence against them. He then immediately turns the table on the bad guys by asking them one sensible question which they fail to answer.
Princess Celestia, who in addition to being a kind, gentle pony (willing to forgive her formerly insanely evil sister after Luna tearfully apologized), is also a ruler that isn't as stuffy and proper as one would think, as she once pulled a light-hearted prank on the Cakes to get them to loosen up during a visit from her. Two episodes in particular highlight this trait for the Princess:
"A Bird in the Hoof", in which her reaction to Fluttershy's well-intentioned bird-napping of her pet to nurse it back to health is primarily to chide Fluttershy for not asking about the bird, a phoenix on the tail end of its rebirth/death cycle, in a "Could Have Avoided This!" Plot, in addition to having said phoenix apologize to Fluttershy for intentionally making her life difficult when the pegasus was only trying to care for it.
"The Return of Harmony" has the ponies fall into Discord's illusions and all of them but Twilight get corrupted into their opposite element. When Twilight herself falls under and is about to leave, the Princess's response is to send back all the friendship reports she wrote her; this causes Twilight to be freed from the Mind Rape and realize everything, and frees her friends so they can defeat Discord.
"Lesson Zero" has Twilight Sparkle having a monumental Freak Out because she hadn't written her weekly aesop report to Celestia. The princess shows up after Twilight's attempts to rectify the situation have wreaked chaos upon Ponyville, and is implied to given Twilight a well-earned, offscreen lecture before starting to reassure her. When her friends burst in begging Celestia to forgive Twilight, citing their not taking her feelings seriously as the cause of the problems, she takes the chance to drive the lesson home and lighten Twilight's (perceived) workload by having the others join in the reporting only when there's something to report.
In "The Crystal Empire," Celestia gives Twilight very specific instructions for how to resolve the episode's crisis... which Twilight then throws out once the situation gets truly bad. Although Celestia is seemingly disappointed that things didn't go the way she wanted them to, she praises Twilight for having the good sense to recognize the severity of the situation and make an appropriate judgement call, instead of blindly following orders.
Rarity is this for Sweetie Belle in "Ponyville Confidential." Rarity confronts Sweetie Belle upon learning that her little sister is one-third of the new local gossip columnist. Despite her anger (since a recent story had been excerpts from Rarity's diary), Rarity scolds her, uses her own going through Sweetie Belle's stuff for proof to drive home the point about personal privacy, and finish it up by questioning whether Sweetie Belle wants writing hurtful gossip to be her destiny-declaring Cutie Mark. Everypony else was either so angry they sent the fillies away rather than risk an out of line outburst or just shunned them outright.
In the episode "Sweet and Elite" we have Fancypants, a wealthy and influential unicorn that befriends Rarity. Unlike many other high society ponies, he's humble, kind, and by the end of the episode establishes himself as both a Nice Guy and Uncle Pennybags when he stands up for Rarity and her friends in front of what is basically the cream of Canterlot high society.
Twilight herself is this for Spike, being the main adult figure in his life. She pulls him back when he starts to go out of line.
From the children's TV series Recess, we have Principal Prickly, who, although tough on disobedience and often somewhat at odds with the main characters, has joined many an Enemy Mine or cut them a very generous amount of slack when he felt their hearts were in the right place.
Not to mention King Bob, the "ruler of the playground".
Don't forget the kids' teacher, Mrs. Grotke. Probably the antithesis of the Sadist Teacher trope.
Although playground monitor Ms. Finster is generally shown as a stern authority figure, she is usually portrayed as unfailingly fair and given several humanizing episodes.
Really, most of the adults in Recess are portrayed this way, with the exception of Smug Snake Mayor Fitzhugh. Substitute teacher Mr. E is another textbook example.
Scooby-Doo! Mystery Incorporated: Mayor Nettles in season two who, unlike the previous mayor and most of the adults in general, is generally supportive of Mystery Inc.'s actions. In fact she actively worked to get the group back together in the season two premiere.
The Simpsons has plenty of unreasonable authority figures (strangulation is a running gag), but some are this, including of all things the Commandant of the military school in The Secret War of Lisa Simpson. When the Simpson family observes a group of cadets standing at attention, Homer mocks them by throwing stones and the Commandant seems to be genuinely shocked: "They're just children, Mr. Simpson!"
Judge Roy Snyder is also this. Although he has his quirks (the clown on his desk) he is actually one of the few authority figures in Springfield who can't be bribed and is pretty lenient in his punishments.
When he's not yelling at Skinner, Superintendent Chalmers can be fairly reasonable and was perhaps the only teacher able to reach out to Bart.
Played with in "Homer Goes to College" where Homer is made to enter a college nuclear physics course to be eligible to work in the power plant. He imagines the dean to be a severe authoritarian like the Animal House parodies he's watched, and never seems to realise the dean is really a friendly and fair-minded person.
Of the four rulers in Shadow Raiders, King Cryos of Ice is far and away the most sensible. It's a lucky thing that he's the first one Graveheart and Tekla have to deal with.
Cortes in Skyland, though he often wonders why he listens to kids.
South Park had Chef (until they killed him off). It wasn't so much that nobody else would believe the kids, so much as he was often the only vaguely competent and intelligent adult in the entire town. It tends to be a town full of Mulders, with the kids usually playing Scully. Also, Principal Victoria and President Bush, as he was the only one in "Cartoon Wars Part 2" who defended the Family Guy writers rights to free speech instead of having them arrested like many people were recommending.
The Governor of Colorado in Cartman's Silly Hate Crime 2000, who was willing to pardon Cartman after the boys pointed out the hypocrisy of charging him with a hate crime.
King Kenric is the only authority figure in Super 4 who actually plays the trope straight, while Dr X has too much of an ego to listen opinions, the Fairy Queen exiled Twinkle due to mistakenly turning her into a frog, and Captain Sharkbeard is outright antagonistic.
Teenage Mutant Ninja Turtles (2003): In "Sons of the Silent Age," the nuclear power plant foreman reads his men the riot act for not noticing the buildup of toxic waste in the dike that was about to spill into the river. He also declines to pursue the "environmental do-gooders" who blow up one of the abandoned buildings to reinforce that dike, recognizing that their actions prevented a disaster.
In Teenage Mutant Ninja Turtles (2012), Splinter is this. It's highlit in the season one episode "It Came From The Depths", where he professes that Michelangelo is right when he says that the Turtles owe it to Leatherhead to give him a chance, but also agrees that Leatherhead must be reasonably restrained, since he is a super-strong giant gator-man prone to sporadic bursts of violent fury.
Sir Topham Hatt from Thomas the Tank Engine often fills this role toward the engines whenever something flies off the handle. Best examplified in "Hero of the Rails" after Thomas lets him in on Hiro — instead of scrapping him or sending him away, he declares they must help him at once.
Transformers Animated has Ultra Magnus from Cybertron (who unfortunately has Sentinel Prime as his primary advisor), and Captain Fanzone from Earth. Later in the series, when Ultra Magnus is out of action and Sentinel Prime takes over, Alpha Trion fills this role.
Transformers: Prime has Special Agent William Fowler. While he does get frustrated with the Autobots from time to time, he lets them do their thing as long as no one gets injured (besides Decepticons and MECH). And when the Autobots need something only a human can provide, Fowler will cooperate if he's available.
In TRON: Uprising there's Able, the boss of Able's Garage. He stands up to the Occupation whenever they harass his employees, and lets Beck get away with having so many breaks and being absent so often due to Able knowing that Beck is The Renegade.
Faragonda, the headmistress of Alfea in Winx Club. She is generally kind to her students, though she is willing to put up with no nonsense from them, and will discipline them as she sees fit. As well, she does not take kindly from students from other schools causing trouble at Alfea either and will take action if necessary.
In the Justice League Unlimited episode "Hawk and Dove," the North Kasnian leader might qualify. Both sides of the civil war are being manipulated by Ares, who just gets a kick out of seeing people kill each other, and to this end supplies the North Kasnians with a superweapon called the Annihilator. Ares is furious when the North Kasnian leader calls off his troops once they drive the South Kasnians off their land since he doesn't want to run the country, he just doesn't want his people to be oppressed. When Ares insists that he use the Annihilator to wipe his enemies off the face of the earth, insulting him in the process, the North Kasnian leader immediately breaks off their agreement and tells him to bugger off and take the Annihilator with him. Unfortunately, Ares (probably) kills him right after and impersonates him to lead the attack himself.
Melanie Baker from Clarence is the most (and possibly only) reasonable faculty member at Clarence's school.
Samurai Jack: In "Jack Remembers The Past", Jack recalls bullies picking on him and refusing to give him his ball. When he tells his father, his father is sympathetic, but he says he cannot interfere with the bullying because he did not want to abuse his power. In the end, he trusts his son to learn how to handle them...which he does.
Queen Angella is this for the Rebellion. She agrees to give Adora a fair chance, despite her history as a Horde soldier, though she's understandably wary. She compromises by having her daughter Glimmer (who vouched for Adora) be her handler. She eventually realizes Adora can be trusted, and gradually gives her and Glimmer more responsibility and praise.
Hordak is a rare example of a villain being this, and a lowkey Benevolent Boss, to boot. He rewards competency and determination, and punishes those who refuse to admit to their mistakes, but not in the usual, lethal You Have Failed Me sense. Even when Catra's attempt at attacking the Rebellion at the end of Season One fails, he still promotes her, because her track record up until then is incredible, and she did come way closer than any of her predecessors did. He also doesn't get angry when Entrapta talks back to him, since her argument was a genuinely fair point.
Unfortunately, as of season four, all authority figures listed are either dead or mind-wiped.
In the SpongeBob SquarePants: Mr. Krabs sometimes fills this role toward SpongeBob sometimes when he's not being greedy.
First exemplified in "Pickles" when he sees SpongeBob confused due to forgetting the pickles and helps him get his confidence back.
In the infamous "A Pal For Gary", Mr. Krabs oddly fills this role compared to other episodes, as he understands SpongeBob needs to tend to Gary and is happy to let him work at the Krusty Krab. And rather than insist he finish the parking lot that instant, he allows him to come in five hours earlier for the sake of the snail. He's also oddly kinder than usual compared to how he acts later in the season.
Done again in "Yours, Mine and Mine" when he punishes SpongeBob and Patrick for fighting over Patty Pal and immediately resolves their feud.