I first heard of fractured couplets when reading the works of Richard Armour, but I image they started when the first caveman who wrote a poem showed it to another caveman who was the first wiseass. It’s so easy a caveman can do it!
Tthe basic rule is to quote the first line of a song and add your own humourous rhyming second line; i.e.
Suzanne takes you down to her place by the river,
Where she’ll have a meal of fava beans, Chiati and your liver.