Mash the potatoes in a bowl with the butter or margarine, then mix in the tuna, parsley, salt and pepper to taste and half of the beaten egg. Cover the mixture and chill in the refrigerator for 20 minutes.
Place the tuna mixture on a floured surface and shape into a roll. Cut into 8 slices and shape each one into a flat round, about 6 cm/2/½ inches in diameter. Dip into the remaining egg, then coat with breadcrumbs.
Heat the oil in a frying pan, add the fish cakes and fry for 2-3 minutes on each side or until golden brown and heated through. Serve with a tomato salad.