Using a box grater, coarsely grate the potatoes. Wrap in a clean tea towel and squeeze out the excess liquid over the sink. Transfer to a bowl and stir in the onion, rosemary and salt and pepper.
Heat half the oil in a large frying pan. Divide the potato mixture into quarters and spoon into 4 x 12 cm (5 inch) mounds in the pan, pressing down to form patties. Cook over a medium heat for 5 minutes on each side, transfer to warmed serving plates and keep warm in a moderate oven.
Heat the remaining oil in the frying pan for about 1 minute until very hot, add the eggs, 2 at a time, and fry until the whites are bubbly and crisp looking. Serve the eggs on the rösti, garnished with chopped parsley.