Actually, he doesn't have to pay anything, but he always asks the original artist's permission before he does the parody, and his band records a new accompanying music track (if he sampled the original music, then he would probably have to pay). A running Weird Al joke is that he continually takes shots at Prince for never giving him permission to parody his songs. If someone wanted to re-record a Weird Al song, would the eight cents per copy go to Weird Al, or would four go to the original composer?
Obviously Weird Al would only have to pay if the original composer demanded payment, but if the original composer refused Weird Al would not be allowed to use the music at all (unless the music were rewritten and parodied).
All to Al.