Avatar: The Abridged Series is, like its many brethren, at heart a fan series that isn't afraid to make friendly jabs at the source material. While much of its humour is derived by changing aspects of the original show, a lot of it still comes for exaggerating character traits and pointing out the show's plot holes and illogical-ness. For example, Sokka points out some anachronisms in episode 10:
Sokka: Let me get this straight. You can invent tanks (invented 1915), jet skis (1973), and a gigantic freakin' drill (20xx). But the concept of a hot air balloon (1783) eluuuuuuudes you. The Mechanist: Umm... yes. Sokka: I hate this world and everyone in it.
Twilight Sparkle: The best parodies are those which positively build on the original work, rather than rely on things like gratuitous cursing and unrelated pop culture jokes. There's a reason the original show caught people's interest in the first place, and paying tribute to that isn't a cop-out... IT'S COMEDY!
Yu-Gi-Oh! The Abridged Series proved this in Episode 50 with an immensely encouraging speech from Joey about copyright infringement but at its core showed how affectionate the series really is.
Joey: Yeah, I have a closing statement. Maybe we have committed copyright infringement. But you gotta know we've done everything in our power to support the Yu-Gi-Oh! franchise! And if it weren't for us, I don't think the show would be nearly as strong as it is right now! Johnson: And where is your evidence of such? Joey: Look around, Johnson! There are more Yu-Gi-Oh fans now than ever before! And the more you try to stifle our creativity, the more we'll try to express our love for a show that's about more than children's card games! It's about fighting for what you believe in, and I believe in this show and its fans now more than ever! Because they believe in me!
Star Wars Uncut is a parody remake of Star Wars using 15 second videos from people all over the world. Some are more faithful than others, but they parody it because they love it.
Superman And Man has the Pre-Crisis Superman watching Superman: The Movie. That scene lampshades the way the 1987 film is even cornier and sillier than the Pre-Crisis comics it's based on; yet still it's acknowledged the fact that Superman is an entertaining movie made with care.
This YouTube series of fan videos is a parody of the whole superhero genre. Each episode is a therapy session of various heroes and villains from the the DC or Marvel Universe (plus Darth Vader and Palpatine in the last one), looking at their various psychological issues.
"Two-Face Therapy!" also introduces a parody of Dr. Phil replacing temporarily the usual therapist.
The author's favorite game show, Interceptor, is also affectionately parodied in The Interceptor.
A Series of Stories is a parody and satire of the recurring cliches in the My Little Pony: Friendship Is Magic fandom. The writer is a passionate brony and loves the fandom, and just wanted to poke fun at the cliches the fandom created. He even makes it a point to include cliches he likes just to ensure it stays that way.
Equestria: A History Revealed parodies all of the headcanons and the unknown nature of Equestrian history, by creating a completely absurd version of Equestrian history that somehow, fits completely within canon. It also lovingly parodies academic essays in general, and fics that are written in an essay-like style, by blowing out any semblances of academic professionalism and logic out of the water while maintaining its own essay-like structure, with a ridiculously crazy Lemony Narrator and her equally crazy conspiracy theories.
Pokémon Reset Bloodlines has one sidestory that acts to the now-dubbed Resetverse, the Dude Gaiden. It also includes several scenes that parody specific oneshots written by the different authors.
Skyhold Academy Yearbook invokes this in-universe when the kids at the eponymous school form a Drama Club and put on a School Play version of Beauty and the Beast (2017). Before beginning final dress rehearsals, director Dorian allows them to do one run-through which is purely silly, to get the giggles out of their systems. The way they treat the dialogue, and even one of the songs, is clearly this.