Preheat oven to 200ºF. Cut skate fillets into 2 equal pieces. Season with salt and pepper on one side. Dredge in flour on both sides, shaking off excess.
Heat a large nonstick skillet over medium flame. Add 1 tablespoon each of oil and butter, swirling to mix as butter melts. Place 2 skate fillets in pan, seasoned side down. Sauté for about 3 minutes, then carefully turn with a wide spatula. Cook on second side until just cooked through, 2 to 3 minutes. Transfer fillets to plate and keep warm in oven while you cook the remaining two fillets, first adding more oil and butter to skillet, if necessary. Transfer second batch of fillets to oven too.
Wipe skillet clean and return to flame. Melt remaining 4 tablespoons of butter in skillet. Cook, occasionally stirring, until butter browns, 3 to 4 minutes. Meanwhile, plate skate fillets. Add lemon juice, zest, capers and parsley to skillet, stirring. The lemon juice will make the pan spatter like crazy. That's okay. Just be careful. Turn off the heat and spoon sauce over fillets. Serve immediately.