Remove the avocado pulp and reserve the shells. In a bowl, mix the avocado with lump crab, chopped celery, and onion. Season with salt and pepper, and chill for 30 minutes.
For the lime citronette, blend lime juice with salt, white pepper, and olive oil until smooth. Stuff the avocado shells with crab mixture and top with citronette, diced tomatoes, and parsley.