I think the distinction between parody and satire is that the latter has to offer some sort of commentary. A parody can just be goofing on something, presenting it in a funny or absurd way, while a satire offers some insight into its object which forces us to consider it in a different light.
Space Balls parodies science-fiction, but
Blazing Saddles satirises Westerns; watching the former shouldn't necessarily change the experience of watching science-fiction, but the latter should make you think differently about Westerns. It's not a clear line, of course, because sometimes simplistic goofing can be sufficient to constitute satire; a poster of a dictator with a funny mustache drawn on become satirical because it forces you to think of the dictator not as an iron superman but as just some dude. The distinction I think has to be one of purpose rather than of high-mindedness, or we end up with something equivalent to the old chesnut that you like porn but I like erotica.
What they have in common, I think, what distinguishes them from simple mockery, is that they must in essence use the object against itself. They have to draw their essential humour from the terms of their object, and while parody is obviously able to play a bit looser with this because it's going for the laugh rather than the point, but it should still draw out some contradiction, tension or absurdity in the original. If you re-write Star Trek as a workplace comedy, it's not a parody (at least, not of Star Trek), because Start Trek isn't a workplace comedy. Replacing the bridge crew with
foul-mouthed Taysiders is a parody, because it's taken something present in the original, turned it a bit, and the humour comes from watching it fall apart. ("I'ss laik hee-haw wiv ivver sin afor, cap'n, ken.")