Since nobody so far seems to have answered the copyright question, here goes.
Parodies, answer songs, and suchlike are all examples of "derivative works", and as such, the original copyright holder is entitled to a proportion of the royalties.
A composer also has "moral rights" to a composition. That means that if, for example, Adge Cutler and the Worzels record Combine Harvester, Melnie Safka can insist on being identified as having written the tune, since it is a parody of "Brand New Key".
Rarely, a writer can invoke moral rights to prevent the release of a parody or derivative which they find morally objectional.
So I could sue someone who added a homophobic rap to my song "Watch What You do in the Night"- I would too!
Cheers
Dave