The trouble with using non trad tunes for parodies is that you are supposed to get permission to use the tune for that parody. If the composer doesn't want you to parody his/her tune or words, you can get stung with a very hefty bill, for breach of copyright..... Most are OK about it, but some can get monumentally stroppy about it, mentioning no names (streets of London....).

Les Barker's credits get absurd after a while, crediting the original tune and composer, the original words and author, the parody words, the 'adapted' tune, etc...

LTS