Add the potatoes to a large pot and pour in cold water until they're covered by about an inch. Once the water boils, let them cook about 30 minutes, until a fork slides into the center with no resistance.
While they cook, heat the oven to 425°F. Brush a large rimmed baking sheet with about 1 teaspoon of the olive oil, or give it a light coat of cooking spray.
Drain the potatoes in a colander. Leave them there for 5 to 10 minutes so the steam escapes and the skins dry.
Arrange the potatoes on the baking sheet in a single layer, with a few inches between them. Steady each one with your hand and press it flat with the bottom of a measuring cup, a potato masher, a large fork, or a spatula. Thinner potatoes brown more; thicker ones stay softer in the middle.
Spoon the 3 tablespoons melted butter and the rest of the olive oil over the potatoes. Season evenly with 1 teaspoon salt, ½ teaspoon pepper, and ¼ teaspoon garlic powder.
Bake for 30 to 40 minutes, until the edges are deep brown and crisp.
Scatter the ½ cup Parmesan and the parsley over the hot potatoes right away so the cheese melts. Taste one and add a pinch more salt if needed.