Scrape the potatoes or leave the skins on, if you prefer. Cut them into thick slices. Chop the onion.
Heat the oil in a large frying pan, add the potatoes and onion and stew gently, partially covered, for 30 minutes, stirring occasionally until the potatoes are softened. Strain the potatoes and onions through a colander into a large bowl (set the strained oil aside).
Beat the eggs separately, then stir into the potatoes with the parsley and plenty of salt and pepper. Heat a little of the strained oil in a smaller pan. Tip everything into the pan and cook on a moderate heat, using a spatula to shape the omelette into a cushion.
When almost set, invert on a plate and slide back into the pan and cook a few more minutes. Invert twice more, cooking the omelette briefly each time and pressing the edges to keep the cushion shape. Slide on to a plate and cool for 10 minutes before serving.