Well, when you think of an AMV you think of anime and music, while generally when you think of a MAD, it's set to VN's and original drawings; also there are MMVs, which are, well, made with manga. But the definition of AMVs includes MMVs and what we usually think of when talking about MADs.
However, MADs and AMVs aren't entirely overlapping. MADs also include things which are not part of AMVs, which include a lot of the M@Ds based on commercial spots and/or movie/real life thing audio mixes and so on.
For example, those pesky Ronald McDonald videos are M@Ds (
http://www.youtube.com/watch?v=lI5YFVagu8M ), as well as all those M@Ds that "remix" an audio/video feature to sound like a song, most often some DDR or Beatmania ( some to red zone:
http://www.youtube.com/watch?v=vtfpybWWCso http://www.youtube.com/watch?v=9oTYo8kD6U8 http://www.youtube.com/watch?v=3pFM_PX9xI4 ). These videos are not necessarily AMVs, since they do not always use anime as the source (when they do, they count as amv, so for example, this is also an AMV:
http://www.youtube.com/watch?v=MDKXd6965fg ), but when they use sources which are not fine for AMVs, well, they are M@Ds, yet not AMVs.
On the other side of the coin, to my better knowledger, trailer videos (at least the ones set to actual trailer audio), which are not too rarely done as AMVs, are not considered MADs at all. So there we go.
MADs and AMVs are for the most two intersecting sets, but they each have their own differences. You could be editing a video that falls under both definitions, or you could be doing one that doesn't.
Re: OP Parody: Scintilla's Eva Bebop, for example. But either way, OP Parodies do count as AMVs as long as the source used was an anime/manga/game/visual novel. The differences come in when the visual source is not one of those, as far as op parodies are concerned.