You can perform anything you like, at least in the US. Copyright issues are the concern of the venue. You only need to worry if you record or otherwise distribute, or if you are the venue -- like, if you rent the hall and organize the show yourself.

As for recording parodies, I seem to remember hearing about cases going both ways. It might be safer to get permission from the copyright holder, even if it means paying a few cents per copy of your CD. If you are making money on it, the original writer of the song deserves some compensation anyway.

John Peekstok