Cut the lemon in half. Save one half, squeeze out the juice and keep aside until ready to make the sauce. Take the other half and cut in 4 wedges to serve along with the fish when ready to serve. Season the fillets lightly with salt and pepper, dip in the milk, and dredge them in the flour. Shake off any excess flour.
Heat 4 tablespoons of butter in a large skillet over medium-high heat. When the butter stop foaming, add the fillets. Cook turning once, until golden brown and fully cooked. About 5 minutes. Transfer to the dinner plates and keep warm.
Lower the heat under the skillet and swirl in the remaining 2 tablespoons butter. When the butter is melted, but not foaming, add the juice of the half lemon and the fresh parley. Sprinkle lightly with salt and pepper to taste. Remove from the heat. Spoon the sauce over the fillets and serve the lemon wedge along side the fish.