1. Bring a large pot of salted water to a boil. Add potatoes and cook until tender but still firm, about 15 minutes. Drain and set in the refrigerator to cool.
2. In a mixing bowl, combine mayonnaise, dill pickle relish, mustard, salt, and pepper. Mix well.
3. Add the cooled potatoes, celery, red onion, and parsley to the bowl. Mix gently until all ingredients are evenly coated.
4. Chill in the refrigerator for at least 1 hour before serving.