Season flour to taste with salt and pepper; dredge fillets in flour mixture, shaking off excess.
Heat oil with 2 tablespoons of the butter in large nonstick skillet over medium heat. Add fillets without overlapping (cook in 2 batches if needed).
Cook until golden on bottom, about 3 minutes. Turn fillets over and cook 1 to 2 minutes more or until opaque through thickest part.
Drain fillets on paper towels; transfer to platter and keep warm. Add remaining 4 tablespoons butter to skillet and heat until it froths.
Add lemon juice and lemon slices; swirl pan to heat through. Pour butter sauce over fillets immediately before serving. Sprinkle with parsley, if desired.
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