Parody mocks an original piece of work by using humorous, satiric or ironic imitation. Satire itself is much broader.
Some examples: Swift's A Modest Proposal, some of Mark Twain's stuff, etc.
The Daily Show with John Stewart, The Colbert Report, The Simpsons, and the Onion can also sometimes be examples (although sometimes those productions are just plain silly).
Sometimes, satire or parody will show up on newspapers or blog. Every time that happens, readers, not understanding that what they're reading is satire, will post angry (or sometimes, sadly, complimentary) comments. It's clear from their comments that they just didn't get that it was satire.
Why is satire so often misunderstood? Has it always been misunderstood? Did the people in Swift's time really think he advocated eating children?
In any case, it's a powerful way of showing the ridiculousness of certain positions in a powerful way...but only if the reader/viewer understands that it's satire.