Heat the olive oil in a 10-inch skillet over medium-high heat for 1 minute. Reduce the heat to medium and add the potato slices a few at a time so they don't stick together. Add the onions and cook, stirring occasionally, until the potatoes are tender and the onions are golden but not brown, about 10 minutes. Place a strainer over a bowl in your sink and transfer the potatoes and onions to drain; reserve the oil.
Whisk the eggs, milk, salt and pepper together in a large bowl. Drain excess oil off of the potato mixture. Add the potatoes and onions to the egg mixture and combine until they are completely mixed in. Set the bowl aside for about 15 minutes to let the potatoes release some of their starches into the eggs.
Heat 3 tablespoons of the reserved olive oil in a medium nonstick skillet over medium-high for 1 minute. Add the egg-potato mixture, rotating the skillet in a circular motion to distribute it evenly. Lower the heat to medium-low and shake the pan a few times to prevent sticking (by shaking the pan, you ensure that the eggs and potatoes release from the pan bottom). Cook the tortilla for about 5 minutes or until the potatoes on the bottom start to turn golden brown. Place the skillet in the oven and cook for 6 to 8 minutes, or until almost set. Remove from the oven and place a large flat plate on top of the skillet and invert the skillet. The tortilla should come right out. Add a tablespoon of the reserved olive oil to the skillet and slide the tortilla back in to cook the other side until it is golden-brown, about 5 minutes.
Turn the heat off and set the skillet aside until the tortilla cools to room temperature. Transfer the tortilla to a plate sprinkle with parsley and serve at room temperature cut into triangles or squares.
*Cook's Note: The Yukon potatoes should be sliced using a mandolin.