Put a large pot of water on high heat. Add the potatoes and salt and bring to a boil. Boil just until a skewer or knife can easily penetrate the center of the potatoes, 20 to 30 minutes. (see How to test potatoes for doneness).
When the potatoes are done, draw off about 1 cup of the cooking liquid; set aside. Drain the potatoes and return them to the pot in which they were cooked. Mash them with a potato masher. With a wooden spoon, stir in the olive oil. Add some of the cooking liquid or milk until you reach the desired consistency. Season generously with salt and several grinds of black pepper.
Just before serving, check the consistency of the potatoes and add a little of the cooking liquid or milk if they need loosening. Mix in the parsley. Taste and adjust seasonings.