Light a grill or preheat a grill pan. Season the fish cavities with salt and pepper. Stuff each cavity with a thyme sprig, a bay leaf and 2 lemon slices. Rub the outside of the branzino with the olive oil and season with salt and pepper.
Grill the branzino over high heat, turning once, until browned and crisp and just cooked through, about 7 minutes per side. Serve right away, passing salt, lemon wedges and parsley at the table.