Rinse the fish inside and out and pat dry. Place the fish and shellfish in a shallow plate and sprinkle on all sides with lemon juice. Drizzle olive oil over everything and season inside and out with salt and pepper. Stuff the whole fish with a few thyme sprigs.
When the grill is too hot to hold your hand 6-inches above the grill for more than a few seconds, it's ready. Grill the fish without moving it until golden brown grill marks appear. Turn it over and finish grilling on the second side; until the flesh is moist and opaque. Some fish may take longer than others; check often and remove the fish as it finishes cooking.
To serve, arrange the cooked fish on a large serving platter and drizzle all over with extra-virgin olive oil. Mix any leftover lemon juice with olive oil and coarse salt to make a dipping sauce. Garnish with the parsley and lemon halves.