2 Live Crew won their case when they did a raunchy cover of Roy Orbison's "(O) Pretty Woman." I don't think they had to pay licensing fees, nor does Yankovic.
EDIT: I asked about licensing fees, and Wiki[1] says that Coolio accepted royalty payments for "Amish Paradise." So I still wonder how 2 Live Crew settled their suit, and I need to get back to work.
Using the music verbatim would not likely be fair use, and the videomaker owes a license fee.