The reason that I believe this parody is derived from Oh, No John because of the verse from that song that goes "May I tie your garter...".  

There are four versions of Oh, No John in the Cecil Sharp manuscripts (see Idiom of the People pgs 33-37) and all of them contain the "Tie your garter..." verse -- even if the song as collected was only two verses it still retained the "Tie your garter...".    Here is a version of Oh, No John from the Sharp mss:

Madam shall I tie your garter
Tie it a little above our knee
Suppose my hand should slip a little farther
Wouldn't you think it rude of me?
No Sir No, No Sir No.
Still the fair maid answered No

My love an I we went to bed together
There we lay till the cocks did crow
Unclose your arms my dearest jewel
Unclose your arms and let me go
No Sir etc.

[MS # 1457.  Sung by Alfred Emery (age 78) at Othery in 1908]