Preheat the oven to 400 degrees F. In a large non-stick ovenproof saute pan, over medium heat, add the oil. When the oil is hot, add the onions. Season with salt and pepper. Saute until caramelized, about 8 minutes. Add the sausage and saute for 2 minutes. Add the potatoes. Season with salt and pepper. Continue to saute for 1 minute. In a mixing bowl, whisk the eggs until frothy. Season with salt and pepper. Using the back of a spoon, lightly press the chorizo/potato mixture over the bottom of the pan. Pour in the egg mixture.
Place in the oven and bake until the center is firm to the touch, about 15 to 20 minutes. Remove the pan from the oven and cool slightly. Slice into individual servings and place on serving plates. Garnish with parsley.