Using a sharp knife, cut a small sliver from the bottom of each tomato so it makes the bottom flat enough for them to stand upright without rolling over.
Cut the top ¼ off the tomatoes. Scoop out the seeds and pulp and discard (be careful not to scoop right through the bottom of the tomato). Set tomatoes aside.
In a large bowl, combine the crabmeat, mayonnaise, onion, celery, red pepper, parsley, chives, salt and pepper. Mix until thoroughly combined. Cover and refrigerate until ready to use.
In a small bowl, combine avocado, sour cream and salt and mash with a fork until smooth.
To assemble: Spoon a small scoop of avocado mixture into the bottom of each hollowed out tomato. Top with a spoonful of the crabmeat and garnish with chives.