Melt the butter in a large saucepan over medium heat. Add the onion and cook until softened, about 5 minutes.
Add the potatoes and stock, and season with salt and pepper. Add the bay leaf and bring to a boil. Reduce the heat and simmer until the potatoes are tender, about 15 minutes.
Add the haddock to the soup and simmer for 5 minutes, until cooked through. Remove the bay leaf.
Using a slotted spoon, transfer the haddock and potatoes to a bowl. Set aside.
Add the milk to the soup and simmer for 5 minutes. Return the haddock and potatoes to the soup and heat through.
Stir in the parsley and serve.
Interesting Facts
Cullen skink is a traditional soup from the town of Cullen, Scotland.
The soup is traditionally served as a starter or light meal.
Cullen skink is often served with a side of crusty bread.