Place potatoes in a pot with water to cover. Bring to a boil, then reduce heat, cover and simmer 25 to 30 minutes until potatoes are fork tender. Drain, rinse under cold water, and drain once more.

Cut potatoes into 1 inch pieces. Place them in a bowl, pour in vinegar, and add salt and pepper. Add chopped celery and cornichons. Stir in mayonnaise and mustard until everything is combined. Chill, covered, until ready to serve. Sprinkle with dried parsley and a dash of paprika before serving.