Trim ends of each tomato and cut crosswise into 1/2- inch slices. Cut each avocado half lengthwise into four slices.
Place a tomato slice in the center of each of 12 chilled salad plates. Fan 2 avocado slices over each tomato slice and season lightly with salt and freshly ground black pepper, to taste.
Whisk 1 cup Spicy Vinaigrette in a large mixing bowl, add crabmeat and gently combine, keeping lumps intact.
In a separate bowl, whisk remaining vinaigrette and drizzle over each serving of tomato and avocado. Spoon a portion of the crabmeat over the edge of the tomato and avocado, allowing most of it to fall onto the bare plate, and decorate with a parsley sprig. Serve immediately.
Combine all ingredients in a medium nonreactive mixing bowl, and whisk until well blended. Cover and chill overnight.
Chef’s Notes: If the oil congeals, return the vinaigrette to room temperature and whisk vigorously immediately before using.