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.
When almost set, invert on a plate and slide back into the pan and cook a few more minutes.
Invert twice more if you need to, 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.