The Austin Lounge Lizards have always been one of my favorite funny bands. I love "Hey Little Minivan" and "Another Stupid Song About Texas" (being originally from Texas).
Weird Al is always hilarious. Well, not always. But most of the time. I've always liked "You Don't Love me Anymore": Why did you disconnect the brakes on my car?/ That kind of thing is hard to ignore/ Got a funny feeling/you don't love me anymore.
My all time favorite funny folk song, though, is Andy Breckman's "Railroad Bill." My brother plays and sings it and I die laughing.