In an 8-inch skillet that's 1 1/2 inches deep, heat two-thirds of the olive oil over medium heat. When oil is hot, add potatoes, and cook for 10 minutes. Add onions, season with salt, and cook for 10 minutes more. Stir in chorizo, garlic, and parsley, being careful not to crush potatoes; cook 2 minutes. Transfer to a medium bowl to cool slightly.
Lightly oil a large plate; set aside. Wipe skillet clean using paper towels; heat remaining olive oil over medium heat. Season eggs with salt and pepper. Using a spoon, gently mix eggs into cooled potato mixture; add mixture to pan. Cook until eggs are half cooked, gently pulling edge of tortilla toward the center so any uncooked liquid runs underneath. Let sit, untouched, over low heat until underside of tortilla is almost cooked, 2 to 3 minutes. Slide tortilla onto oiled plate, and invert back into pan; cook until both sides are cooked and the center is still soft. Slide tortilla onto a serving plate. Serve warm or at room temperature.