Place whole potatoes in a large pot and cover with water. Bring to a boil and cook until tender when pierced with a fork, about 20 minutes. Drain and set aside to cool.
In a medium bowl add mayonnaise, vinegar, mustards, garlic, salt and pepper. Stir until blended.
Peel and cube potatoes. Place potatoes in a large bowl with onions, celery, eggs and parsley.