Heat the olive oil and one tablespoon of the butter in a large skillet. Meanwhile, quickly season the fillets with salt and pepper and dredge them in the flour. When the oil and butter are hot, lay in the fillets and let them gently pan fry until golden brown, about three minutes over moderate heat. Flip the fillets and brown the other side. Unlike with many other fish, skate’s flavor and texture are best when it cooked through.
Remove the fish to a platter and tent it with foil to keep it warm while you make a quick pan sauce in the same skillet you used for the fillets.
Heat the remaining three tablespoons of butter in the skillet. Add the garlic and let it cook to soften for a couple of minutes. Stir in the white wine and lemon juice and let the mixture bubble, stirring and scraping up the bits of flour and skate left in the pan to make a sauce. Stir in the chopped capers and parsley, and spoon the sauce over the fillets.
Garnish with a fine grating or two of fresh lemon rind.