Put the potatoes in a large saucepan and cover with cold water by 1 inch. Season generously with salt and bring to a boil. Reduce the heat to medium and simmer until very tender, about 15 minutes.
Drain the potatoes and return to the saucepan over low heat; cook until any remaining water evaporates, about 2 minutes. Add all but 1 tablespoon butter; mash with a potato masher until smooth. Add 1/2 teaspoon salt, then gradually mash in the milk until creamy. Transfer to a serving dish; top with the remaining 1 tablespoon butter and chopped parsley.