Place potatoes in a pot with water to cover. Bring to a boil, cover and simmer, stirring to ensure even cooking, until a thin-bladed paring knife or a metal skewer inserted into a potato can be removed with no resistance, 25 to 30 minutes. Drain, rinse under cold water and drain again. Cool slightly.
Cut warm potatoes into 3/4-inch dice with a serrated knife. Layer them in a bowl, seasoning with vinegar, salt and pepper as you go. Cut eggs, celery and pickle in 1/4-inch dice and thinly slice scallions. Add to potatoes, along with parsley. Stir in mayonnaise and mustard until everything is combined. Chill, covered, before serving.
