Wash and peel (peeling is optional) potatoes; slice into 1/4" rounds.

Heat oil in heavy frypan until very hot, but not smoking; fry potatoes for 5 minutes, turning occasionally, until golden-brown; remove potatoes to absorbent paper to drain.

Beat eggs well with parsley; add salt and pepper to taste; add the drained potatoes.

Discard all but 3 tablespoons of the oil from the frypan; reheat if necessary; slowly pour egg-potato mixture into the pan; distribute the potatoes evenly; cover frypan and cook omelet for 4 minutes on medium-low heat; turn omelet over and cook, covered, another 4 minutes; top with some grated cheese, if desired, to melt during the last couple of minutes in frypan; carefully remove omelet from pan and cut into 4 servings.