"Give me your A, said the fiddle to the banjo
Or your E or D, I just don't care."
So we sat around while the band was tuning
All night long, at the Roseville fair.

Definitely poking fun, but not at Mr. Staines or his fine song.And I think that a great deal of punch results from using a familiar song as a framework.

Essentially a good parody is good; a bad one is bad. Much like other songs.