I don't think the word 'parody' is accurate here. A parody usually introduces another theme or element. I think the word 'burlesque' is much better. A burlesque either alters the words slightly in some way to make them comic but retains the same plot and theme, or keeps the song pretty much as was and the burlesque is all in the performance. Of course there are different degrees of this. There are burlesques of many of the more common Child Ballads, like George Collins, Lord Lovell and Lord Randall. I will look into this when I have more time but my point was that apart from the proclaimed burlesque of the 19thc already mentioned the early versions show other hallmarks of burlesque, but this is only my opinion.

Richie,
I don't think Buchan's version is in the Harvard Ms. It's in the BL ms.
It's titled 'Bonny Barbara Allan' FL 'In Scarling town where I was bound' 41v and in the ballad she is referred to as 'Babie Allan'. I didn't have time to copy it out but I have seen it. The last verse mentions a 'Captain Green'. Peter had an extensive library and seems to have drawn on multiple copies for his heavily edited versions. He will have had a copy of the 18thc versions as one was printed in Aberdeen in 1775 (BL 1346 m 7 16b). Child had a copy of Peter's version but chose not to include it. I wonder why!!