Parody is a literary genre whose objective is to mock its subject by retelling it in a way that plays up its shortcomings in a humorous way. It is sometimes considered to be a form of satire, but it depends on who you ask; the two are very similar, and variance in their definitions causes them to overlap more or less depending on which definitions you use. The main difference is that satire indirectly makes fun of its subject, which is usually something broad like society or a type of person, whereas parody points a very clear finger at its subject, which is usually a specific work, event, or individual.
Counterintuitive though it may be, the best parodies tend to be founded in love for the subject material. For instance, the guys who wrote Bored of the Rings had the utmost love and respect for J.R.R. Tolkien. That element of irony is partly why it's so funny.
Parody fic is a type of fanfiction that pokes fun at its source material, another fic or fanwriter, or a common trope in fanfiction. Parody Suefics, crossovers, and slash are not hard to find. Some parody fics purposefully mock a particular badfic or even certain badfic writers.
A parody fic done well can be side-splittingly funny, but if done poorly, it is as bad or occasionally worse than what it is making fun of in the first place.
Claiming, if challenged, that your fic is a parody will not generally get it out of being badfic. If someone had to ask, that tends to mean it was poorly written. Being "so bad it's good" is not the same as being a parody.
Like any other genre, parody can be done poorly, especially if horrible writing is involved. In particular, parodies founded in hate tend not to be very good, since they often devolve into simply bashing the subject and therefore are not funny. Also, if the parody has plainly misunderstood its subject, it will be dumb instead of entertaining—the best humor is based in truth, after all. Tasteless humor (e.g. rape jokes, 9/11 jokes) is another highly effective way to anger people instead of making them laugh.
Trollfic may be considered a type of bad parody in that it imitates real badfic in an attempt to provoke readers. It often gives itself away by a characteristic over-exaggeration of bad spelling, grammar, and logic, not to mention capslock abuse.
PPC missions themselves are parodies of the specific fanfictions being sporked. (This is why they're sort of pseudo-legal.) They may also satirize bad writing, or facets thereof, in general. Missions tend to walk a fine line between love and hate, since PPCers typically love their fandoms and fanfiction in general, and definitely love the canons involved, but despise what the bad fics do to said canons. PPCers are encouraged to focus on finding the humor in the bad writing, and are strongly discouraged from bashing badfic authors.