Peel the potatoes and cook in boiling salted water until just tender, then drain and cool. Cut into quarters lengthways and slice very thinly. Peel and finely chop the onion.
Heat the oil in a frying pan and add the onion. Fry gently until the onion has softened, then add the potato and fry a little more quickly until both are just beginning to brown. Draw the pan off the heat. Crack the eggs into a mixing basin and add a good seasoning of salt and pepper. Add the potato and onion mixture.
Melt the butter in a clean frying pan. When frothing pour in the omelette mixture. Stir for a moment, then allow the omelette to cook slowly until the underneath has browned and the mixture is set but still moist. Loosen the omelette and turn over on to a hot plate. Slide it back into the pan and cook the second side. Turn out on to a plate and sprinkle with chopped parsley. Serve hot or cold.