Grate the potatoes using the large holes of a box grater. Put into a large bowl of cold water, stir to remove the starch, drain and rinse. Put the grated potatoes in a clean kitchen towel and squeeze to remove excess moisture.
Heat the butter and oil together in a 12-inch nonstick pan over medium-high heat. Season the potatoes with salt and pepper. Place the potatoes in a single layer in the pan and cook until the underside forms a brown crust, 3 to 4 minutes. Begin to stir the potatoes and continue to cook until browned all over and cooked through. One minute before they're done, stir in the parsley and garlic. Serve immediately.