There might be two distinct uses/meanings of the word "parody", which just means "a song sung beside": the ordinary one, a mocking/satirical take--off; and a less common one, a sustained set of allusions to a previous artwork... (See Grove: the word "parody" in music apparently has no sense of mockery attached to it.) In the latter sense, possibly old words to new tunes and new words to old tunes could both meaningfully (usefully?) be called parody? I apologise in advance if such is not the case.

Sanjay Sircar