Peel the onion and chop finely.
Heat the oil in a large saucepan and add the onion. Fry gently over a low to medium heat for 3-4 minutes.
Meanwhile, peel the potatoes and cut into small dice. Add the potato and vegetable stock the the pan, and season well with salt and black pepper. Bring to the boil and simmer for 5 minutes.
Finely chop the parsley stalks, (setting aside the leaves for later). Add the frozen peas and parsley stalks to the pan, then simmer for a further 5 minutes.
Use a slotted spoon to remove about a third of the vegetables to a bowl.
Add the parsley and the juice of half the lemon to the soup, then blitz to a very smooth purée in a blender or using a hand blender in the saucepan.
Tip the reserved vegetables back into the saucepan - these will provide a lovely chunky texture to the otherwise smooth soup. Taste and adjust seasoning, then serve.