As I scrolled through this thread (which I seem to have missed in its earlier appearance) I eventually found that McGrath of Harlow had said what I would have said:

"Most traditional folksongs are new songs based on old songs." (My emphasis)

Indeed, what I would have answered to Mark's original assertion that

"... we need a new name for a song that uses the tune and format (and even some of the words) of another song, but isn't making fun of that song or its theme." is:

A folk song.

I often end up using the word "parody" in real, or verbal, 'quotes' in discussing the folk process ... but I'm trapped by the unfortunate overtones of the narrow modern use of "parody".

Regards,

Bob Bolton