In a large mixing bowl, combine the mayonnaise, Dijon mustard, lemon juice, and Old Bay seasoning. Mix well until smooth.
Add the shredded crab meat, chopped celery, red onion, and bell pepper to the bowl. Gently fold the ingredients together until everything is well coated with the dressing.
Season with salt and pepper to taste. If desired, add the chopped parsley for an extra burst of flavor and color.
Cover the bowl with plastic wrap and refrigerate for at least 30 minutes to allow the flavors to meld.
Serve the crab salad on a bed of lettuce leaves or in a sandwich, if preferred.