The Capitol Steps satirize and parody both sides of the politcal spectrum in song. They are former Senate staffers (I was also a Senate staffer when they started in the 80s, but I cant sing) who have since turned it into a full time career, recording 20some CDs and touring and performing around the country.
http://www.capsteps.com/
Trent Lott singing "It's Not Easy Bein' White" and the song about domestic wiretapping, " Do You Fear What I Fear?" from their Christmas Bush cd are pretty funny. I'll have to look for the one of Hillary singing "I'll do it my way"
Historically, its interesting that most political satirist seem to be on the progressive side. From Mark Twain and Will Rogers to, more recently, Lenny Bruce, Mort Sahl and Richard Pryor.