Slice the potatoes into thin matchsticks (1/8-inch thick) with a vegetable slicer or mandoline, dropping them into a bowl of cold water as you cut. Drain the potatoes and dry them thoroughly with paper towels. Drop the potatoes in batches into the hot oil and cook for 3 to 5 minutes, until golden brown. Remove from the pot with a wire basket skimmer or slotted spoon and drain on paper towels. Place on a baking sheet, sprinkle with salt, and keep warm in the oven while you cook the rest of the potatoes.
Sprinkle the potatoes with parsley, if desired, and serve hot.