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!
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.