1. Preheat grill to medium-high heat.
2. Score snapper by making 4 (1-inch) crosswise cuts on each side of skin with a sharp knife. Sprinkle 1/4 teaspoon pepper and salt evenly inside cavities; arrange lemon slices evenly in cavities. Wrap fish tightly in foil coated with cooking spray. Place fish on grill rack; grill 15 minutes or until fish flakes easily when tested with a fork, turning once. Let stand 5 minutes before unwrapping.
3. Combine parsley and remaining ingredients in a food processor; process until finely chopped. Stir in remaining 1/4 teaspoon pepper. Serve sauce with fish.