Bring potatoes, bay leaves, garlic and cold, salted water to cover to a boil in a Dutch oven. Simmer until potatoes are tender, about 12 minutes. Drain. Discard bay leaves and garlic.
Combine scallions, celery, vegetable oil, vinegar and mustard in a bowl. Season with salt and pepper. Add potatoes.
Stir in mayonnaise and parsley and serve at room temperature or slightly chilled.