Add the potatoes to a large pot of cold water. Season with salt and bring to a boil. Simmer until tender, about 15 minutes.
Heat the olive oil in a large pot. Add the onions and saute until tender and just golden. Add the milk to heat and season with salt and pepper.
Drain the potatoes well in a colander. Add them back to the pot you cooked them in and mash until smooth. Fold in the hot milk and onions, extra-virgin olive oil and chopped parsley.