"Peasant or country-style." Often a robust, rustic sauce or dish made with pancetta, mushrooms and Parmesan cheese or with root vegetables, particularly onions, and tomatoes most often served with pasta.
"In the style of Parma" in Emilia-Romagna, and not necessarily with Parmesan cheese but may be, or with the famous ham of the neighbourhood. Most commonly applies to veal escalopes dipped in flour, egg and breadcrumbs fried and served with tomato sauce and mozzarella.