BOIL potatoes in a large saucepan, covered with water, 25 to 30 minutes, until fork tender. Rinse with cold water until cooled; drain.
CUT cooled potatoes in 1-inch cubes. Peel if desired.
MIX mayonnaise, celery, onions, mustards, pickle relish and parsley in a large bowl. Fold in potatoes and eggs and mix to combine. Season with salt and pepper to taste.
COVER and chill if not serving at once. Keeps up to 3 days.
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