Bring a large pot of water to a boil. Add 2 tablespoons of salt and the pasta and cook according to the directions on the package. Set aside 1 1/2 cups of the pasta cooking water before you drain the pasta.
Meanwhile, heat the olive oil over medium heat in a pot large enough to hold the pasta, such as a 12-inch saute pan or a large, shallow pot. Add the garlic and cook for 2 minutes, stirring frequently, until it just begins to turn golden on the edges-don't overcook it! Add the red pepper flakes and cook for 30 seconds more. Carefully add the reserved pasta-cooking water to the garlic and oil and bring to a boil. Lower the heat, add 1 teaspoon of salt, and simmer for about 5 minutes, until the liquid is reduced by about a third.
Add the drained pasta to the garlic sauce and toss. Off the heat, add the parsley and Parmesan and toss well. Allow the pasta to rest off the heat for 5 minutes for the sauce to be absorbed. Taste for seasoning and serve warm with extra Parmesan on the side.