1. Place potatoes in a large pot and pour in enough cold water to cover by 1 inch. Add 1 Tbsp. salt and bring to a boil over high heat. Reduce heat to medium-low and cook until potatoes are tender, 13 to 17 minutes. Drain, return potatoes to pot and cover with a damp kitchen towel. Let stand, off heat, until soft, 10 to 15 minutes.
2. In a small bowl, combine mayonnaise, mustard and vinegar. Cut potatoes into halves or quarters. In a large bowl, mix potatoes with celery, shallot and parsley. Fold in mayonnaise mixture and season well with salt and pepper. Serve at room temperature.