1. Rinse the clams under cold water and place them in a bowl. Sprinkle the cornmeal over the clams, add cold water to cover, and let soak for 1 hour. Drain, rinse again under cold water, and drain again.
2. Scrub the mussels well and rinse them in several changes of cold water to remove any sand. Drain. If they have beards, don’t pull them off until right before you cook the mussels.
3. Place a lobster on a cutting board and plunge the tip of a sharp knife about 1½ inches down from the eyes into the center of the head; cut straight through the body and head. Then cut lengthwise completely through the whole tail. Remove and discard the sac in the top of the head, the tomalley (green liver), and the intestine. Carefully crack the claws with the back of the knife. Repeat with the remaining lobsters.
4. Preheat the oven to 475°F.
5. Combine the butter, 4 tablespoons of the parsley, the garlic, and salt and pepper in a small bowl.
6. Divide the lobster halves, cut side up, and the clams, mussels, and shrimp between two large rimmed baking sheets; Season the seafood well with salt and pepper. Spread the butter mixture over the lobsters and shrimp and on the clams and mussels.
7. Bake until the lobsters and shrimp are cooked through (the meat should be opaque) and the clams and mussels have opened, about 20 minutes. Discard any clams or mussels that do not open.
8. Arrange the shellfish on one or two large platters. Sprinkle with the remaining 2 tablespoons parsley and garnish with the lemon halves. Serve immediately.