Soak the pearl barley for 2 hours in plenty of cold water.
In a heavy pan with a lid, fry the carrot, onion and celery in the oil and butter until softened. Add the diced prosciutto and stir in with a wooden spoon. Drain the barley and rinse it, then add the barley to the vegetable and prosciutto mixture, stir for 5 minutes. Add the stock and bay leaf and bring to the boil. Reduce the heat, put the lid on the pan and cook the soup gently for 1½ hours, stirring regularly. Towards the end of the cooking time, check the seasoning and add salt and freshly ground black pepper. Just before serving, stir in the Parmesan cheese and parsley.