Unfortunately, this can be a hypertechnical exercise in hairsplitting. The term used by Sundries Shack was "mock," which I am afraid is both incorrect and imprecise. The critical thing to remember is that a parody doesn't have to be funny; "mock" implies a necessary humorous tinge. By the offered definition, Don Quixote would have been a proper "parody." As I read the case law, though, that's extremely dubious; the "creators" of Amadis of Gaul, Tirante the White, Palmerin of England might still have a cause of action against Cervantes, because his parody extends primarily to matters of style. The subject-matter in Don Quixote is far, far more serious than the fluffy (well, as fluffy as one gets when dealing with edged weapons) adventures of the medieval romanza as it descends from the chanson de geste. That makes Don Quixote far more parallel to The Cat NOT in the Hat, and would probably lead to much the same treatment by the Ninth Circuit. What this says about literary judgment in the legal profession is better left unsaid… but could probably be inferred after reading the opinions that appear under the names of all but a handful of judges.
To put it bluntly: Mockery isn't enough unless it extends beyond mere mockery of style or surface. Instead, to qualify as a protectable parody under the "target parody" doctrine, the mockery must extend to the underlying thematic material. The courts have not been nearly specific enough about this; but that conclusion is the only way to reconcile all of the opinions without throwing up one's hands and crying "For want of a circuit split, certiorari was lost." Just try figuring out why "Dr Juice"'s rather pointed look at the OJ trial is not fair use, but The Wind Done Gone (which has less literary merit!) is, without focusing on thematic materials. The less said about Lo's Diary the better!
You should have noticed by now that there's a word I've been carefully avoiding throughout this discussion; it is actually the most important word, in the literary sense, in the whole discussion; and, of course, legal doctrine says it's not fair game. Satire. In an incredibly ill-founded leap of logic that puts Bob Beamon's effort in Mexico City to shame, copyright law distinguishes parody from satire. Parody can be fair use; satire, however, is not. The trick then becomes distinguishing the two. That many literary scholars consider the two terms to be the opposite of how the courts have come to use them—that is, a parody essentially restricts itself to tone and surface characteristics, whereas a work so limited will fall outside of the "Dr Juice" definition—seems to have escaped much notice. Perhaps the saving grace is that a satiric work that falls close enough to a source to be infringing has at least a decent chance of also qualifying as a parody. However, satire should be fair use (if infringing at all), because the point of satire is a reexpression intended to comment (adversely) upon the original, which is the overarching purpose of the doctrine in the first place. Often, that original is thematic material, rather than particular works; Candide and The Crying of Lot 49 are good examples.
I realize that this seems rather far afield from fan fiction at the moment. Just trust me—it will all connect back up in the end.
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I can see already that I'm going to have to rethread these entries on fan fiction; there are at least three left, and I suspect that there will be more "opportunities" for side trips. Some time in the next couple of days, I suppose I'll have to do something with Warped Weft.