Heat the oil in large nonstick ovenproof or cast iron skillet over medium heat. Fry the potatoes until tender and golden brown. Add the onion and salt and pepper. Continue to cook until the onion is translucent, about 3 to 4 minutes.
Meanwhile, whisk together the eggs, cheese, parsley, and salt and pepper in a large bowl. Add the egg mixture to the potatoes and onions. Cook, shaking the pan and gently moving the mixture from side to side with a rubber spatula as some of the liquid from the top reaches the bottom of the pan.
Cook until the bottom is set and beginning to brown and the top is still loose, 2 to 3 minutes. Flip the eggs by sliding them onto a plate. Then invert and slide the eggs back into the pan, cooked side up. Cook until the eggs are set, 2 to 3 minutes more, shaking the pan often to prevent sticking.
If you prefer, rather than inverting the eggs, you can place the skillet under the broiler for 2 to 3 minutes until the top is set and lightly browned. Unmold onto a plate, cut into wedges, and serve.