Preheat grill to medium-high, adding soaked wood chips to coals or a smoker box.
Split lobster tails lengthwise through the shell, leaving meat attached. Crack shell so the tail sits flat.
Whisk butter, garlic, lemon juice, salt, and pepper. Brush over the lobster meat.
Toss broccoli with olive oil, salt, and pepper.
Grill lobster flesh-side down 4 to 5 minutes, flip, brush with more butter, and grill 3 to 4 minutes more until opaque and firm.
Grill broccoli alongside, tossing occasionally, 6 to 8 minutes until charred and tender-crisp.
Garnish lobster with parsley and serve with lemon wedges and extra butter.