a) As others have said, you don't need the permission of the original artist to do a parody. Although it's probably helpful to give credit to the original song's lyricist / composer, and possibly the record company that owns the copyright.

b) We've had numerous relyricised songs here already - Andrea Lena Dimaggio appears to be our resident 'expert' :)

c) They're currently being actively encouraged via the February Music Motifs Challenge.

 
 
--Ben