There are songs like Great Speckled Bird and Thinking Tonight Of My Blue Eyes and Wild Side Of Life that are certainly not parodies, but do share the same tune. Then there are answer songs like It Wasn't God Who Made Honky Tonk Angels; not really a parody either although based on The Wild Side Of Life.

Homer & Jethro were masters of the parody.
A group called The Frere Brothers from Prince Edward County in Southern Ontario have made some wonderful parodies.
Four Two-Fours of Molson Export is a parody of Four Strong Winds about cross border shopping:
"Think I'll go down to Rochester,
Shopping's good there in the mall
Got some friends that I can go shopping for. . ."
or the one about the Mushroom plant in P.E. county to the tune of I'll Tell Me Ma
"She's big and fat and she's my aunt
And she works down in the mushroom plant.
She picks mushrooms one two three
Got a light on her hat so she can see."