Parodies of music videos done by fans of the artist (and there are quite a lot), some examples being this and this. Although at first glance many of the Key of Awesome music videos seem to be making fun of the artists they parody, they have admitted they love most of the artists that they parody, and sometimes the artist loves them.
The sketch-comedy website LoadingReadyRun use this trope all the time. One of their better-known parodies is CSI:CSI - Internal Investigations. Replaced the discovery of a dead body with the stealing and eating of another person's sandwich.
Another skit along these lines was the Channel 101 skit "Cirque du Soleil — Sex Crimes Investigations", which spoofed both crime procedurals and Cirque in a goofy but fun manner.
Italian Spiderman is an affectionate parody of Italian B-Movies of the 1960s and 70s. The website is even complete with the fictional history of its production company, and details of how the movie was lost and recovered only in the early 2000s.
(To the Indiana Jones theme) "Kiss a Wookiee, kick a droid, fly the Falcon! Through an as-ter-oid, till the Princess! Is annoyed! This is space ships, it's monsters, it's Star Wars, we love it!"
A Very Potter Musical and its sequel are full-length musical parodies of the Harry Potterbooks and movies. The sheer amount of effort that must have gone into its production, the very obscure references, and Word of God guarantees that the writers, cast, and crew are all huge fans.
On that same note, Potter Puppet Pals also falls under this trope; however, watching some of the episodes might fall more towards the 'Parody' and less towards the 'Affectionate.'
The Legend of Neil is a parody of The Legend of Zelda in specific, of video games in general and of the subgenre of animated series where someone from the "real world" gets trapped inside a video game.
The Jerry Seinfeld Program is a series of minute-long mock episodes of Seinfeld, using all the real credits and music and a greenscreen backdrop of Jerry's apartment, but exaggerating the characteristics of the show and characters to ridiculous degrees and applying the Seinfeldian Conversation style to inappropriate situations, like fundamentalist christianity or agoraphobia.
Suparna Galaxy, Leigh Alexander's spoof of Mass Effect. On the one hand, it's a vicious condemnation of the game's reliance on turgid mathematical infodumping and thinly-veiled Dating Sim elements to look meaningful. On the other hand, only a slavish fan of Mass Effect could have written a word of it.
Hilly and Hannah Hindi (of The Hillywood Show) are more or less devoted to musical parodies of Twilight (allfourofthe movies), Pirates of the Carribean, Harry Potter, and other broad targets. Hilly also does a scarily-accurate impression of Lady Gaga.
The Many Adventures Of Lividmanda is a parody of Survivor(in particular the Micronesia season) made by a poster on the snarky Reality TV message board Survivor Sucks. It was quite popular there, and the expression "I'm Livid!" was a minor meme among online fans of Survivor for about a year. Some of the jokes won't make sense if you're not familiar with past seasons of the show or if you're not familiar with Survivor Sucks, but the story can be enjoyed by pretty much anyone if you want a quick laugh.
This flash game is a parody of Little Red Riding Hood where instead of the big bad wolf going after the little girl to eat her, he gets bopped in the head by a piece of candy, sees a trail of candy, and decides to eat all the candy instead. The candy is dropped by a pair of children and the wolf, despite the fact that he could eat the children if he wanted to, hides from the kids when they look his way by pretending to be a bush. If you get caught, the ending (depending on how fat the wolf has gotten) will usually show the wolf meeting his end, such as being beat up by the children, having a bird poke a hole in his bloated belly so he flies around like a deflating balloon, or even being tossed into a pot to be cooked by a witch. Getting the wolf big as possible results in him falling on top of the kids and falling asleep from his candy binge.
While the Whateley Universe is generally parodying superhero comics (when not playing them straight), the story "Tales of the MCO" is a parody of the classic Quinn Martin productions, with a main character who is a parody of Jack Webb in Dragnet.
Two Power Rangers parodies, Mighty Weenie Pathetic Rangers and Post Psychedelic Power People made the rounds in the early 00's. They were close enough to the original scripts (and kept the most important parts as much the same as they could)) that fans could relive memories (this was before YouTube was an option) and diverted enough from the original that fans could have a laugh. The latter continued beyond spoofing Mighty Morphin' Power Rangers and even spoofed Zeo with Power People Zeeto, Turbo with Power People Turkey, In Space with Power People Lost In Space, and Lost Galaxy with Power People Lost Sanity. It's obvious that both creators were devoted fans of the show and based on conversations in newsgroups, admired each others' work. Artist Bros (the creator of the former) even said they were happy that they were the ones continuing beyond spoofing season 3.
Both even spoofed Fox advertisements for other shows with parodies of VR Troopers (VR Boopers), Sailor Moon (Sailor Goon and Sailor Spoon), and other shows that aired on Fox Kids. There were even early drafts for parody scripts.