In a spacious mixing bowl, combine the mayonnaise, lemon juice, and Dijon mustard. Whisk until smooth and creamy, seasoning with salt and black pepper to taste.

Gently fold in the lump crab meat and finely chopped celery into the dressing, being careful not to break the crab meat too much. Add in the parsley and mix evenly.

Cover the bowl with plastic wrap and refrigerate for at least 30 minutes to let the flavors meld.

Serve the salad on a bed of lettuce or in avocado halves, garnished with extra parsley and lemon wedges.