Using a knife, slice each potato crosswise into thin slices, taking care not to cut all the way through. (For even slicing, you can lay 2 wooden spoons or chopsticks along the sides of the potatoes as guides. This keeps the knife from cutting all the way through.)
Place the potatoes in a 9x13-inch or 7x11-inch baking dish and use your hands to carefully fan the potatoes out.
In a small bowl, whisk together the melted butter, olive oil, garlic, salt, and pepper.
Spoon the butter mixture over each potato, making sure to get in the cuts and coat the potatoes well, about 1 tablespoon per potato.
Bake, uncovered, for 45 minutes. Sprinkle the Parmesan and parsley evenly over each potato. Return to the oven and bake until the Parmesan is melted and a little crispy, about 20 minutes. Serve hot.