Boil the potatoes in their skins (or better still, bake them). Peel and mash in the pan while still warm. Beat in the butter and salt. Place the pan over a medium heat and start to add the milk, trickling it down the side of the pan and beating it in with a big spoon. Beat until light and fluffy. Taste and adjust the seasoning. Reheat when necessary.
Before you serve, heat half of the butter and half of the oil in a frying pan over a high heat until the foaming subsides. Fry the black pudding slices on both sides, then transfer to a dish lined with kitchen paper. Add the remaining butter and oil to the pan and fry the apple slices briskly until browned. Drain briefly on kitchen paper, then arrange the apple and black pudding slices over the mashed potatoes. Sprinkle over the parsley and serve.