Pour the flour in a deep plate and season it with some pepper and salt. Coat the fish on both sides with the flour and then shake off any excess flour.
Add the olive oil and a tablespoon of butter to a large non-stick frying pan and place it over medium heat until the melted butter starts to foam. Then add the soles to the pan.
Then remove the soles from the hot pan. Transfer them to a clean plate. Cover with tinfoil. Discard the browned butter that is still in the hot pan. Wipe it clean with kitchen paper. Then add the remaining butter to the pan and place it over medium-low heat until melted. Add the lemon juice and half of the chopped parsley. Season with a dash of pepper and salt.
Take the butter sauce off the heat. Place the fried soles onto clean plates and then drizzle with the lemon butter sauce. Sprinkle with the remaining chopped parsley. Serve hot.