Claude Simpson's The British Broadside Ballad and its Music has a section devoted to Go From My Window, and refers to your parody. The opening and closing sections of the entry are:

""Go From My Window" is named as the tune for singing the fourth section of "Frauncis New Jigge" c 1595 (Pepys... and Baskerville The Elizabethan Jig...; entitled "Mr Attowel's Jigge"...). A ballad called "Goe from the windowe goe" was licenced March 4, 1588, but it is lost unless parts of it are echoed in Merrythought's song in Beaumont's The Knight of the Burning Pestle, III, v: ...
That the original ballad was considerably older than 1588 is suggested but the religious parody in A Compendious Book of Godly and Spiritual Songs, 1567 (ed A.F.Mitchell, 1897, pp132-133)....

Chappell (PMOT I,142) printed a stanza of the old song recovered from oral circulation, with a tune whose opening and closing cadence reminded him of Ophelia's song "How should I your true love know". Within the twentieth century further traditional traces have appeared..."

(If you don't have access to Simpson I could scan the 3 pages and e-mail the entry to you; e-mail me at MCPearce0ATaolDOTcom if you want them).


You might also look at Greg Lindahls Sixteenth Century Ballads - A Work In Progress" to see if there's anything there. Also check out The Traditional Ballad Index and Bruce Olson's site for references.



Mick