Combine the potatoes, onion, olive oil, and salt in a large skillet. Cook over medium-low heat, stirring occasionally, until the potatoes are tender but not browned, about 20 minutes.
Drain the potato mixture through a strainer set over a bowl. Reserve 3 tablespoons of the oil and let the potatoes cool for 5 minutes.
Mix and Set:
Beat the eggs in a large bowl. Fold in the warm potato and onion mixture and let stand for 5 minutes.
Heat the reserved oil in the skillet over medium heat. Add the egg mixture and cook until the edges are set and the bottom is golden, about 5 minutes.
Flip and Finish:
Place a large plate over the skillet. Carefully invert the tortilla onto the plate, then slide it back into the skillet.
Cook until the second side is golden and the center is softly set, 3 to 5 minutes. Rest for 10 minutes, then slice into wedges.
Serve with tomato slices and parsley.
Notes
The potatoes should be tender and pale, not crisp.
Use a plate slightly larger than the skillet for a confident flip.