The particular version in question has been touched upon a few times here in the past, though in the arrangement recorded by 'Ian and Sylvia'.

A Folksong, Captain Woodstock

Captain Woodstock riddle

Lyric, Capt Woodstock's Courtship-Ian & Sylvi (sic)

None of these quote the full text or tune. In the last mentioned, Bruce Olson pointed out that only one traditional version calls him 'Woodstock'; Richard Hartlan's (South-East Passage, Nova Scotia), recorded by Helen Creighton and printed in Songs and Ballads of Nova Scotia (1933), and later in Bronson's Traditional Tunes of the Child Ballads, I, 372-3 (46.22).

A quick search shows no sign of a recording by Peter Paul and Mary, though.