Whirl 3 plum tomatoes, halved and seeds removed, in a blender until smooth. Add 2 tbsp each vodka and clam juice, 2 tsp lime juice and 1/4 tsp each celery salt, Tabasco sauce and Worcestershire sauce. Pour over ice.
Toss halved plum tomatoes with oil. Grill on medium-high until slightly charred, 1 to 2 min per side. Season with salt and fresh pepper. Drizzle with white balsamic vinegar and sprinkle with chopped parsley.