Cut the fins off the fish with scissors. Bone the fish through the belly, keeping both sides attached. Season both inside and out with salt and pepper.
Arrange chorizo, tomatoes, and cheese inside the cavity. Combine butter, garlic, and oregano and spread over both sides of the fish. Fold the fish over and secure with a toothpick. Grill over low heat, until the cheese is melted and the fish is cooked, about 4 to 5 minutes per side. Brush the fish with the wine continuously, using the parsley bunch as the brush.