My father, originally from Blackheath in England's industrial West Midlands, used to add the following to the opening couplet of William Allingham's fairy poem "Up the airy mountain":

Then up jumped a fellow named bottle nose Dick
Said he would do us a conjuring trick
Before you could look or before you could appear
He'd drunk down another bloke's glass of beer
The owner of the beer was so pleased at the joke
He hoped Dick would die of a paralytic stroke
Haul up the anchor pull out the sail
Never tie a knot in a guinea pig's tail
I hope I will remember never to forget
To use an umbrella when it turns out wet

He, and also my mother (originally from Digbeth in downtown Birmingham, England), knew other parodies.

There was this, for example, of "Old King Cole":

Old King Cole was a merry old soul
And a merry old soul was he
He called for a light in the middle of the night
To go to the WC
The moon shone on the closet door
The candle had a fit
Old King Cole fell down the hole,
And stuck to a piece of tape.

(My mother, from whom I learnt this song, although lowly born, was virtuous and well-governed.)

"I Dreamt I Dwelt in Marble Halls" from Balfe's 'Bohemian Girl' (1843) was parodied thus:

I dreamt I was tickling my granddad's bald head
With a piece of oiled cloth (?) and a feathuar
But when I awoke I did find it no joke
He was tanning my a*** with some leathuah.

On Bayley and Bishop's 1830s song "Mistletoe Bough" we had this:

The mistletoe hung on the closet wall
The robbers they came and they stole it all
The old man came out with a great big stick
And knocked them all into a barrel of........
(Chorus) O the mistletoe bough aw aw
         O what a miserable row.