Actually, I suppose "Steepleford Town" can be considered a right wing folk song, since it speaks well of the fox hunting gentry. And The Methody Parson and The Ranter, because they make fun of religious dissenters as sex-minded, thieving, hypocrites.
I suspect there must have been English folk songs mocking the Puritans, as after the antics of the Puritans under Cromwell (abolishing Christmas, stabling horses in the cathedrals, showing extreme religious intolerance) that could well be expected. I seem to recall that the words to the dance, Cuckolds All In A Row (aka Cuckolds All Awry), mock Londoners as cuckolds. (London supported Parliament over the king.) Charles II loved dancing to that tune and he may have had the lyrics in mind.