The Adventures of Super Mario Bros. 3 contains examples of:
All Your Powers Combined: In "Super Koopa", Koopa is able to combine the Mario's powerups, something they can't do. At one point, he uses Raccoon, Fire, and Kuribo's Shoe all at once.
As Long as It Sounds Foreign: Parodied in "The Venice Menace" in which the Italian language is made up of nothing but names of famous Italians and Italian-Americans. Hilariously, one of the names uttered is "Lou Albano", AKA Mario's old actor/voice actor (he and Danny Wells were replaced by the time of this cartoon).
Asshole Victim: Junior from "Misadventures in Babysitting."
Dub Name Change: The writers made up their own names for the Koopalings, called the Koopa Kids in the actual show, before the English version of The Manual was final. That, and some of the references Nintendo ended up using would likely go over most kids' heads anyway.
Easily Conquered World: Earth itself. Several episodes feature King Koopa taking over a country or a city with minimum effort. One even has each of the Koopalings taking over the continents, with nothing more than their magic wands.
Episode Title Card: Done in the style of the title screen for Super Mario Bros. 3 (a curtain raising), naturally.
Expressive Mask: The eyes on the frog suit open and close at the same time as the wearer.
Fanservice: Princess Toadstool wears a swimsuit in more than one episode. Sure, it started as a Beach Episode, but at one point, she gets captured wearing it and spends the whole rest of the episode clad as such.
There's also her Totally Radical '80s outfits in the episode "Kootie Pie Rocks".
Fantastic Racism: Exploited by the Koopas in "True Colors", who paint the townsfolk red and blue and spark riots between them.
Mario: You're so stubborn now that you're red, Luigi! Luigi: Yeah? Well, you're a blue know-it-all!
Foot Focus: A barefoot Princess Toadstool in "Reign Storm".
Full Name Ultimatum: Used on King Koopa in three separate episodes (twice in one of them, too!). Also, interestingly, the only times the name "Bowser" was ever used in DiC's cartoons.
Haven't You Seen X Before?: "Hey, what're you staring at? Haven't you ever seen a reptile ready to conquer the world?!"
Humans Are Ugly: Kootie Pie's reaction to her human disguise in "The Beauty of Kootie".
"I Know You're In There Somewhere" Fight: Played with. When Mario was brainwashed as mentioned above, Luigi tries to remind him of the good times they shared in order to snap him out of it, but this doesn't work. However, when Luigi then plays a prank on him as he had done earlier in the show while they were arguing, (specifically, Luigi pulls on Mario's hat, then his overals, and then his stache',) he snaps out of it.
Left the Background Music On: Cheatsy dances to the underground music on a Walkman until Koopa takes away the headphones.
Leitmotif: Since the show re-arranges most of the music from the game, the tunes are indicative of the character or type of environment that is currently on-screen. Examples: Water Land's theme playing any time there's a swimming scene; Dark Land's theme playing whenever King Koopa or his fortress is on-screen; the airship theme whenever Koopa's airship appears. And appropriately, most episodes ended with the credits theme from the game.
Lolicon: Subverted in "The Beauty of Kootie": Mario uses Kootie's wand to transform Luigi into "Casanova Koopa" to distract Kootie while Mario, Toadstool, the King of Desert Land, and Toad escape. He also kisses her on the lips passionately, which disgusts Luigi himself while in disguise. And later when Luigi reverts back to his former self by Kootie's wand accidentally:
Cheatsy: Luigi!
Luigi: Luigi?
Kootie: Ahh! I've been kissed by a plumber! [Kootie and Cheatsy chases Luigi up the stairs] I'll tie your lips behind your head! And you'll never kiss again!!!
MacGuffin: The wand in "A Toadally Magical Adventure."
Magic Music: The Doom Dancer Music Box made anyone who could hear it dance uncontrollably. This became a plot point as Koopa, who had a cold, couldn't hear it and thus wasn't affected when Bully tried it on him.
Meaningful Name: All of the Koopa Kid's names for this show. Some (such as Bully, Big Mouth and Cheatsy) are more blatant.
Mermaid Problem: In one episode Mario was rescued by a mermaid named Holly Mackerel, who fell in love with him. Complicating matters was the fact that she was a reverse mermaid, and that she saw him in his Frog Suit, which made her think he was a frog.
Moral Dissonance: In the episode "7 Continents for 7 Koopas," Bully Koopa grafittis the Great Wall of China. How does Mario solve this? He turns the wall into a dragon. That's more of a problem than what Bully originally did to it.
Near Villain Victory: Koopa came THIS close to winning in "Never Koop A Koopa", with everyone captured and at the mercy of Bob-Ombs. He only lost cause Cheatsy decided to cheat him.
It seems they were intentionally mocking this trope, since when Koopa sends Kooky to Hawaii to make sure the Princess stays there, all he gives him to fit in with the humans is a lei to wear around his neck and it works.
This doesn't work for the good guys. In one episode the Mario Bros. try to disguise themselves as sledge brothers to stow away on the Doomship. Luigi is worried that it won't work, Mario tries to assure him that he barely recognizes him. But once they get aboard, an actual sledge brother deadpans "Puttin' on a little weight, aren'tcha Mario?"
Portrait Painting Peephole: Demonstrated in "Never Koop a Koopa" when Koopa watches Princess badmouth him through the eyes of his own portrait.
Shown Their Work: This show was the closest adaptation of the games, even featuring P-Wings and the Kuribo's Shoe in an episode, as well as the ruler of Desert Land.
The music box in "Do the Koopa" may reference a similar item from the game.
Super Power Meltdown: Toad and Princess Toadstool use a filled-to-the-brim chest of power-ups that Toad well-stocked in "Super Koopa" to keep Mario and Luigi powered up each time King Koopa de-powers them, though they sometimes cause Koopa to be de-powered. Both sides use their power sources (a pendant for King Koopa, the chest for Mario and Luigi) to keep the fight going. The constant stress of this, however, causes King Koopa's pendant to short-circuit, and we see him turn into many powered-up King-Koopa versionsbefore he finally explodes into his normal de-powered self, the pendant useless as he's sent through a manhole into the Mushroom Kingdom again.
Took a Level in Badass: Princess Toadstool took a more active role in the adventures and was less prone to kidnapping than in the Super Show. She and Toad were even able to use power-ups in the same way as the Mario Bros.
Up the Real Rabbit Hole: In typical fashion, Earth is usually referred to as "the real world".
Big Mouth: It's curtains for you creeps! You're history! You've had it! You're done for! It's all over! We win! Time's up! Your clock is cleaned! The sun has set! The fat lady has sung! The cheese has molded! The Super Mario Bros. are no more!
Water Is Air: In "The Ugly Mermaid", King Koopa tries to make a kingdom of fish surrender to him by flooding their air-filled dome with water, to which they react negatively for some reason.
The merpeople even have to wear water-filled globes over their heads so they can breathe while in the city, Seriously, what?