Combine potatoes, onion, and salt in a bowl. Set aside.
Heat oil in a large frying pan over high heat until very hot but not smoking. Spoon potato mixture into oil, spreading evenly. Cook until at least half the potatoes are browned, using a slotted metal spoon to turn potatoes occasionally, 20 minutes.
Set a strainer over a bowl and pour in potatoes; let drain. Reserve 1 tbsp. oil.
Fold eggs into potato mixture in a large bowl. Stir in parsley.
Heat reserved 1 tbsp. oil in a 10-in. nonstick frying pan over medium heat. Pour egg mixture into pan; spread evenly. Cook, lifting edges to check color. When eggs are medium brown, remove from heat. Set a large plate upside down over pan. Invert tortilla onto plate, then slide back into pan. Cook over medium heat until bottom is browned, about 4 minutes.
Transfer to a plate; cut into wedges.
Note: Nutritional information is per appetizer serving.