Chop the parsley finely. Salt and pepper the soles inside and outside.
Quickly roll them in flour and tap to remove excess flour.
In 2 large frying pans, heat a third of the butter with the oil until the butter browns a little.
Fry 2 soles at the same time in two separate pans until golden, 5 minutes on each side, starting with the white skin side first. Keep basting with the butter.
Once the soles are cooked, place them in a serving dish, pour the lemon juice over them and sprinkle with parsley.
Melt the rest of the butter in a frying pan and, once it turns brown, pour it over the soles.