Heat oven to 400° F. Heat 1 tablespoon of the oil in a large ovenproof skillet over medium heat. Add the yellow onion and cook for 5 minutes.
Add the chorizo, potatoes, and ½ teaspoon each salt and pepper and cook, covered, stirring occasionally, until the potatoes are tender, 10 minutes.
Stir in the parsley. Pour in the eggs and stir to distribute the ingredients. Sprinkle with the cheese and transfer to oven.
Bake the omelet until puffed and brown around the edges and a knife comes out clean, about 15 minutes.
Divide the lettuce and red onion among plates and drizzle with the remaining oil. Cut the omelet into wedges and serve with the salad.