Some parodies become more famous than the originals — Lewis Carroll springs to mind here: his "You are old, Father William" has completely outlived Southey's original.

A folk example:— I just started singing Sydney Carter's brilliant "A Reel of Recording Tape' to myself as I read this, and honestly took about 5 minutes to remember the name or words of the song it parodied - "The Dundee Weaver". Not that that is a bad song; but Sydney Carter's is better, sharper, more satirical — in fact more memorable altogether.

However, my first reaction on reading the OP was, "How could anyone bring themselves to parody Willy McBride? Surely there must be some limits to filthy taste." I hope no-one has done The Band Played Waltzing Matilda. If they have, I do not wish to hear it, thank you!