In a medium bowl, stir together crab, mayonnaise, parsley, onion, and sugar until well combined. Add salt and pepper to taste, then chill until ready to serve.
Arrange lettuce on a large platter. Place about 2 tbsp. crab mixture onto center of each leaf. Top each with a slice of avocado, about 1/8 tsp. cocktail sauce, a lemon wedge, and salt and pepper to taste.
Make ahead: Prepare through step 1 and chill, covered, up to 1 day.
Note: Nutritional analysis is per serving.