There’s a fine line between hommage and parody. Some of the best parodies today are mash-ups, but I think that falls a bit outside the intent of this thread. Here are a couple contenders. Flight of the Conchords has already been mentioned.
Although better classified as a cover artist, I believe the net takeaway of the music of Richard Cheese is one of parody. Classic stuff…
Just discovered the YouTube page of Freddy Wickham, who writes exquisite classical music parodies in the Anna Russell/Peter Schickele tradition:
What about Andy Sandburg’s Lonely Island crew from SNL? They’re entire shtick is absurd songs earnestly performed in contemporary musical styles.
What makes them even funnier is they often collaborate with artists like Akon, T Pain, Charli XCX, Michael Boulton, and Justine Timberlake. parodying their own musical styles..
NOTE: Corrected spelling.
Both Andy Samberg and Michael Bolton have said, in separate interviews, that when The Lonely Island wrote a song about Michael (and not yet knowing if Michael would be up for it), it was ten thousand times more vulgar than the final product (which is still pretty vulgar). Bolton had them tone it down considerably. “I’ve performed for popes!” he said.