I understand 'parody' to mean a song that uses the same tune and often the same structure of the original to poke fun at SOMETHING. Occasionally that something is the original song, especially if it is a slow serious one (like Grey Funnel Line). I think the force that motivates the parodist is like that of a punster; he or she mishears the original, or some synapse suddenly goes snap, and different words come out.
But to look at the original complaint (Grey Flannel Line) the force that creates a parody seems also to be the part of our human nature that says, "Enough seriousness, already, lighten up, let's be silly."
Different people hit that place at different times (between 3 and 5 a.m. at an all night song circle, all the totally ridiculous songs emmerge blinking into the lamplight*), and some people never hit it (as far as I can tell).I once mortally offended a male friend by singing a bawdy parody of White Coral Bells, and in his estimation ruined the song for him forever.
For me, to work, a parody has to be sharp and witty; incisive, not just similar to the original; Grey Flannel line doesn't work particularly well for me.
Then there's the frequency problem. In my experience the parody shows up when I've heard way too many renditions of the original. But the parody's shelf life is considerably shorter. Once or twice is usually enough.
Parodies that delight me: No Moggy Land (nails TWO of Eric Bogle's songs in one, and does wonderfully on that "... again and again and again and again" line. Call Me the Whale (its answer to the Greenland whalefishers) Irving Abramovitz (We tried the other day to sing Spencer the Rover and no one could remember the right words; Irving is too close).
So. Parody serves a purpose, it can be and is overdone, and it takes a strong original to survive a good parody.
My $.02 worth.
Blessings,
Barbara

*Songs like The George Washington Bridge Song; Lloyd George knows my Father; The fish He is a Funny Fish, and Why Does the Winkle always turn to the right?