On the difference between parody and burlesque as applied to the material we are discussing: Some thoughts.

Burlesque exists mainly to ridicule the original piece and is suggesting this is faulty in some way, is already an obvious target for ridicule. Often there is little obvious difference between the original and the burlesque. In fact, as one man's meat ...., burlesque can and does easily become a serious song again so that this and the burlesque can happily exist side by side in oral tradition though not in the same repertoire, but certainly in the same printer's stock and sometimes even printed together on the same sheet.

Parody uses the original as a vehicle and creates a new work, whilst acknowledging the original to some degree. It can take the form of satirising the original but this is not a necessary component. If the original disappears off the radar whilst the parody becomes popular, the parody aspect disappears eventually.

I might be wrong on this one but parody seems to require a single work as its target, whereas burlesque can be attacking a whole genre.

The following to me are pure burlesque
Oh Cruel were my parients (Oh Cruel)
Villikins (William and Dinah)
Billy Taylor (Bold William Taylor)
Lord Lovel
Botany Bay (Cockney)
Giles Collins
Two sisters (Child Vol 4, p448)
Barbara Allen The Cruel
Molly the Betrayed (Gosport Tragedy)
Georgy Barnwell
Sam Small(Sam Hall)
Ah(Oh) My Love's dead (The Lover's Lament for her Sailor)

Here are some obvious parodies of similar songs

Joe Muggins (Lord Lovell)
Giles Scroggins Ghost (Giles Collins)
The Vorkhouse Boy (Mistletoe Bough)

I'm not suggesting there is a hard and fast line to be drawn between the two and I'm sure there are cases which overlap. For instance it could be said that there are elements of parody in Billy Taylor.