Heat 1 tablespoon oil over moderate heat in each of 2 large saute pans until hot.
Put flour into a shallow dish and season with salt and pepper. Dredge fillets in flour, shaking off any excess flour. Dip the fillets back into the milk and then again into the flour.
Place fillets in saute pans and cook over medium heat for 2 to 3 minutes per side, until golden brown and crispy.
Transfer fillets from pans to a platter, sprinkle with parsley.