Dick,

I don't claim any expertise, but I think variation would be a better description than parody. "Do me ama" is almost certainly the older variant and is described on the cover notes of Blow boys blow as being derived from the chapbook tale "The squire and the Farm servant". The twist of the seaman outwitting the gentleman, and of the lady being taken with his audacity makes the story rather more interesting, and I would guess more popular at the time.