There are really two kinds of parody. One takes the whatever out of the song itself, the classic case being the Billy Connolly "D.I.V.O.R.C.E" take on the Tammy Wynette song. Also the Athenry one quoted.

The other just uses a well known song as an audience will be familiar with it and applies it to a different.

As with all genres there are good, bad and ugly - Les Barker is probably the finest current exponent of the parody. "Stand by your Van" - being another TW classic used for the situation when waiting for the AA man.