Thoroughly wash the parsley then remove and discard the stems. Chop the remainder coarsely. Scrape the potatoes and cut into large dice. Peel and finely chop the onion.
Melt the butter in a large saucepan. Add the onion, cover with a lid and fry gently without browning for a few moments until the onion is soft. Add the potatoes and chopped parsley and fry for a further few moments over low heat. Stir in the stock and the milk. Season with salt and pepper and bring up to the boil. Cover with a lid and simmer gently for 1 hour. Draw the pan off the heat and purée in a blender.
Stir in the cream, check the seasoning and if necessary thin down with a little extra stock. Chill well before serving.