Heat a large skillet over medium-high heat. Place the branzino fillets on paper towels and dry well. With a sharp knife, score the skin of each fillet by making three shallow cuts on a slight diagonal. Sprinkle the fillets on all sides with the salt.
Add the oil to the hot pan; then add the fillets, skin side down, being sure to lay the fish away from you in the pan. Press gently on each fillet to make sure the skin has full contact with the pan. Allow the fish to cook undisturbed for 3 minutes. Using a fish spatula, gently flip the fillets and cook for another 30 seconds. Remove the branzino to a platter, skin side up, and keep warm.
To the hot pan, add the tomatoes, capers and olives. Cook for 1 minute, stirring with a wooden spoon. Add the white wine, reduce the heat to medium, and cook until the wine is reduced by half, about 2 minutes. Stir in the butter and parsley. Spoon the sauce over the branzino and serve.