Aquila: One kid freaks out about their mother dating a new man, and spies on him, convinced he's up to something. Eventually he gets used to it.
Arrested Development: This TV show played with this trope whenever Michael got a new love interest. He always worried about what his son thought of his dating, and usually ended up sabotaging the relationship because of that. Of course, while George Michael would usually be fine with it, the two times he didn't consult him were the two times he would in fact have minded. One, when Michael started dating the teacher he had a crush on, after misinterpreting George Michael's opinion of her. And two, with Jesse, because she's psycho.
Bones: Booth's son Parker doesn't like his mother's current boyfriend, who appears to be a decent guy. Of course, Booth doesn't like said boyfriend either. But also inverted, as Parker seems to like Bones, his father's work partner and potential (very potential) love interest. But then, Parker idolizes his dad (and mimics his dad's sarcastic nickname for mom's boyfriend) and Bones has a pool she lets him come over and swim in (Parker at one point suggests his dad should marry just for this reason, being largely unaware of their UST). In later seasons, Brennan has had Booth's baby and they live together, Parker treats her just like a regular mom and seems happy when he spends time with them.
One episode played with it by making it seem that Parker was unhappy that not only were Booth and Bones living together, but having a daughter. Parker was lying to both of them about where he's spending his afternoons, breaking the new radio controlled car Booth got for him, and destroying pictures of Bones. In the end, it turned out that their relentless worrying ruined his surprise. He'd spent his afternoons making a mobile out of all their family photos and motorized it using the parts from the car so that his new little sister would know her family (and especially her big brother).
The Brady Bunch: This is the case of Parent With New Paramour taken to the ideal conclusion, with a relative minimum of tension between the parents and step children.
Buffy had her mother dating Ted, a gentleman whose only flaw was being a psychotically domineering killer robot. Even before she knew this, Buffy responds to a suggestion of her mother getting married again by threatening to kill herself.
There's Giles, who had a somewhat tentative but eventually friendly and entirely mature and responsible platonic friendship with Joyce. You know, except for that one time they ate the band candy, reverted to the mental age of 16 and had sex on top of a police car. Twice. Something of a Subverted Trope there, since Buffy canonically thinks of Giles as a father, but reacted with horror when she found out about the tryst - though that might have been because she got the gory detailsdirectly from Joyce's brain, and the memory was apparently a very pleasant one. Also Dracula, though that lasted even less time, and was offscreen. She seemed happy after when Joyce went out with a nice guy in season five, Joyce's hilariously squicky jokes notwithstanding. Of course, this being a Joss Whedon show, Joyce died the next day.
Burn Notice: Michael often lets his clients stay at his mother's house. He is hilariously horrified, in one episode, when he finds out she slept with one of them.
Call the Midwife: Timmy actively ships his dad with Sister Bernadette, going so far as to propose on his dad's behalf.
California Dreams: In this '90s teen show, The Ditz Tiffany has been living with her dad since her mom walked out on them years before the time of the show. When Dad starts dating again, Tiffany is fine with it at first...until he starts blowing off his plans with her to be with the new girlfriend. She feels left out and unwanted, and makes a fool of herself trying to prove that the girlfriend isn't on the level (she's actually quite a nice person). The episode ends with something of a Broken Aesop, because while Tiffany apologizes for being insecure and acting like a bit of a bratty teenager, her father never apologizes for ditching her or hurting her feelings.
Californication: Both Karen and Hank date lots of people, and Karen especially gets serious with them. Their daughter Becca has to deal with them and she mostly has no problems with them when the respective relationship is steady, but she would very much like if her parents were back together. Hank and Karen are kind of that constant on-again-off-again couple.
Series/Castle: Alexis, who is used to basically being the mature one in the household, is personally fine with her father hooking up with Beckett.
Dexter: Astor and Cody are very fond of Dexter, their mom's boyfriend, and do not like it when Rita dumps Dexter. They're delighted when she decides to take him back.
Dynasty: Much of the first season focused on this, as the plot is kickstarted when the former Sexy Secretary of the male lead becomes his second wife as well as the female lead, which his adult/teenage sons and daughters 'really dislike. And then the ex-wife shows up in the second season...
Everybody Loves Raymond: In one episode, Debra's mom, who recently divorced her dad, reveals that her dad plans to bring a date to Thanksgiving dinner. Her mom is okay with it, although Debra is upset, even though Ray points out that as long as her mother is happy, she should be happy. Everyone assumes that her dad's date will be a much younger woman, but it is actually a much older woman.
Dr. Andy Brown is a widower; the death of his wife and the family dealing with their grief kicked off the series. His young daughter Delia doesn't like when he dates Linda Abbott, and his son Ephram disapproves also, but mainly because she's ill and he's afraid that Delia will become attached and they will have to go through it yet again. At the end of the series, Andy gets together with his friendly neighbour Nina. Both Ephram and Delia have always liked her, so they will probably be okay with it.
When divorced Nina dated Jake, another newcomer and a young handsome doctor, her son Sam seemed to like him.
Edna, a Cool Old Lady married Irving Harper, who was in turn a Cool Old Guy. It was a wonderful case of December-December Romance and a Happily Married couple. In a flashback, the audience finds out that Edna's son Harold was against the marriage, which hurt Edna's feelings. Irving's children were fairly estranged from him anyway, but were no fans either.
Freaks and Geeks: Has Bill's mom, Susan Ivanova dating his gym coach, Biff Tannen. Not only is this a to-the-letter case of example three, but Bill uses the opportunity to slack off in gym class and mouth of to the coach, placing said coach in a very difficult situation.
Glee: Has Kurt's father Burt finding love with Finn's mother Carole. While they're one big happy family now, at the beginning both boys had issues with the relationship. Finn was afraid that Burt would try to replace his father, who died in Iraq when Finn was a baby, while Kurt jealously (and erroneously) believed that his father would rather have Finn as his son.
Blair is excited to meet her mother's new boyfriend Cyrus, believing him to be a dashing, charming, worldly man. He turns out to be tiny, nearly bald and, well, played by Wallace Shawn.
Blair: I was expecting Cary Grant and I got Danny DeVito!
But, she does learn to at least respect him by the end of the episode. By the time he becomes her stepfather they have formed a pretty close relationship.
Chuck and Lily may also count. Since Chuck's mother died when he was born it's never a case of him feeling like she's replacing his birth mother. They get along great at first, then he hates her for a while after his father dies, but then agrees to let her adopt him. She's practically a better mother to him now than she is to her biological children.
Mad Men: Gives us something of a combination of an Inverted Trope and Type 1: Don Draper invites his secretary Megan Calvet on a vacation to California with him and his kids specifically because she handled them so well on a visit to the office. It's only on the trip that he sleeps with her and then proposes after she's established a firm rapport with Sally and Bobby: when he announces his decision to marry her, they are actually kind of happy it happened.
Mighty Morphin' Power Rangers: Season 3 had an episode with Kimberly fretting about her mother's new boyfriend, a french artist (Justified Trope in that the artist wished for them to move to France with him, putting Kimberly's Ranger career to an end).
Misfits: Nathan and his mum's live-in boyfriend Jeremy do not get on - it seems to be mostly Nathan's fault though, and apparently he has a history of trying to ruin his mum's relationships (he even falsely accused one of her previous boyfriends of sexually assaulting him). He verbally abuses Jeremy a lot, even punching him for no good reason at one point and breaking the guy's nose. Nathan's mum gets sick of the situation and actually kicks her son out of the house, making him homeless. They do reconcile in the end though, and Nathan accepts that he was wrong to stand in the way of his mum's happiness and he moves out of the house permanently.
Pushing Daisies: Has a tweaked example: Ned's father trots him off to boarding school and ignores him in favor of his new wife and kids.
Revolution: Ben with Maggie. Ben's daughter Charlie had a difficult time accepting it in the pilot. However, Charlie came to accept her to the point that she was quite upset with Maggie's death in "The Plague Dogs".
Royal Pains: Hank Lawson was none too pleased when his dad showed up, and was even less pleased when he took up with one of his best clients, "New Parts" Newburg.
The Sarah Jane Adventures: The title character's adopted son Luke, despite initial nervousness (and spying on his mother before finding out that she was disappearing to go on dates), gets along well with her new boyfriend/fiance and becomes the strongest supporter of her new relationship. He even brushes off his friends' valid concerns about the man's empty house with a warning not to spoil his mum's big day. Of course, the relationship doesn't work out; the man had died weeks ago (thus the empty house), and the Trickster had revived him to serve as a trap for Sarah Jane. When he finds out, he chooses to die as he should have.
Some Girls: The main reason Viva can't stand her father Rob's new girlfriend and eventual wife Anna is that Anna is also Viva's sadistic gym teacher and football coach. (Viva also first discovered the relationship when she walked in to find Anna riding Rob.)
Supernatural: When Dean settles down with Lisa for a year, he gets along great with her 11-year-old son Ben. Ben met Dean for the first time when he was 8 years old, and Dean wondered if Ben was actually his son since they had similar behavioral patterns (which was neither confirmed nor denied).
Veronica Mars: Twice with Veronica's father Keith Mars. The first is an attempted relationship with her guidance councelor, but he decides to call it off because Veronica is still coping with the loss of her mother after she abandoned them both. The second is with her best friend Wallace's mother, which lasts quite a bit longer. This one just weirds Veronica out for a while but she later warms to the idea when she sees how they're happy together.