In a deep heavy pot or a deep fryer, add the oil and heat to 325 degrees F.
Cut the potatoes lengthwise into 1/4-inch sticks. Put them in a bowl of cold water and let soak for 15 minutes. Drain and dry them well.
Carefully drop the potatoes, in batches, in the hot oil. Cook until golden brown and cooked through, about 5 minutes. Drain well on brown paper. Let cool.
Increase the oil temperature to 350 degrees F. When you are ready to serve, fry the potatoes again in the hotter oil until golden brown and crispy, 3 to 5 minutes. Drain well on brown paper. Meanwhile, in a small pot, melt the butter with the garlic.
In a large bowl, toss a large handful of fries with some of the garlic butter. Sprinkle with the Parmesan, parsley, and salt while hot. Serve immediately.