Known by this name at least since Turner's Herbal of 1562 (OED), the word may derive from 'care'.

Image: http://www.shipleyhealthstore.co.uk
Original Receipt adapted from Ken Bentley Speciality Foods, 'The Carlin Pea King', DriffieldINGREDIENTS
½lb (200g) carlin peas
Pinch of salt
1oz (25g) of beef dripping (or butter)
Some soft brown sugar (optional and not usually used in the North East)
A splash of rum (optional)
COOKING INSTRUCTIONS
Soak the carlin peas in cold water overnight. Drain and put them in a saucepan of boiling water with salt. Boil for approximately 20 minutes, or until cooked but not overdone or mushy.
Melt the beef dripping (or butter) in a frying pan, drain the carlins and then add them to the pan and fry for two to three minutes.
Serve hot with salt and pepper. Or you can leave them to cool for a tasty nibble later. Alternatively, while still hot, you can add the brown sugar and a drop or rum.
Carlin Sunday is followed by farting Monday, according to East Yorkshire folk lore
One tradition has it that when Newcastle was besieged by the Scots, the citizens might have starved but for the arrival of a cargo of dried peas from Norway on the Sunday before Palm Sunday, known as Carlin Sunday. Tradition, though, is uncertain as to whether that was the siege of 1327 under Robert the Bruce or the Civil War one of 1644, while a very similar story has been applied to Blackpool. Other traditions trace the practice to ancient funeral rites or early church philanthropy, as explained in 1836...

Westmorland Gazette - Saturday 26 March 1836
'A Month in Yorkshire' by Walter White (1861) tells of "the ancient custom prevalent along the eastern coast from Humber to Tyne, had eaten fried peas on Carlin Sunday - Mid-Lent of the calendar"
The compendium 'Notes and Queries' for 1864 has; "On Carlin Sunday grey peas fried in butter called Carlins are eaten. In the public houses customers are treated to their fill of these dainties, highly seasoned with pepper and salt, doubtless with an eye to the consequent increased consumption of beer &c."

Carling Sunday 'reminders' are a regular feature of 19th Century newspapers in the North East, this from the Daily Gazette for Middlesbrough - Wednesday 18 March 1885