Peel potatoes, then cut into 1/4-inch-thick slices with slicer. Rinse in 2 or 3 changes of cold water until water runs clear. Drain and pat very dry.
Heat fat in a 12-inch nonstick skillet over medium heat until melted. Cook potatoes with 3/4 teaspoon salt, turning gently, until coated with fat. Reduce heat to medium-low and cook, turning occasionally, until golden in spots and potatoes are tender, about 20 to 25 minutes.
Meanwhile, finely chop parsley and garlic together.