Remove the black skin from the soles. Chop the parsley sprigs, discard the stems. Season fillets with salt and pepper.
Spread the flour in a plate. Dredge fillets in flour, shaking off any excess flour.
Melt 3 tablespoons of butter in a large skillet. Add a sole fillet or if the skillet is large enough, place 2 fillets at the same time. Cook over high heat for 5 minutes. Turn the fillet and cook on the other side for 5 minutes again.
Set aside and keep fillets warm. Sprinkle with lemon juice and parsley. Cook the other sole fillets the same way. Add butter if needed.
Melt the remaining butter in the skillet. When brown, remove from heat and place the sole fillets.