Preheat oven to 400°F. Scrub the potatoes. With a paring knife, remove skin and about 1/8 inch of the flesh in long 1- to 2-inch-wide strips. (Reserve peeled potatoes for another use.) Toss potato skins with oil, paprika, salt and cayenne. Place in a single layer on a lightly oiled baking sheet; sprinkle with Parmesan. Bake until tender and golden, 25 to 30 minutes.