Well, it turns out that legally recording original lyrics to copyrighted music is the same as recording a "cover" of the song. I'm looking into obtaining permission to use the background music for my three (yes, one more than you may have been aware of) parodies.
The big advantage of using someone else's music is that the song is already a hit, and people know the melody. The down side is that I would have to pay royalties on each and every copy that I give away or sell.
The big advantage of writing my own music/lyrics is that I would own all the rights to the songs and wouldn't have to pay anyone else for the privilege of using their stuff. I would also be free to distribute them as I see fit. The big downside is coming up with all that material...lol.
The point may be moot, as I've cut the songs that I've been thinking about and will have to dream up more lyrics. We'll see...