Adjust oven rack to lower-middle position, set large heatproof serving bowl on rack and heat oven to 200 degrees.
Bring 4 quarts water to rolling boil, covered, in large Dutch oven or stockpot. Add pasta and 1 1/2 teaspoons table salt to boiling water, stir to separate pasta, cover, and cook until al dente; reserve 1/3 cup pasta cooking water and drain pasta.
While water is heating, combine 3 tablespoons oil, 3 tablespoons garlic, and 1/2 teaspoon table salt or sea salt flakes in heavy-bottomed nonstick 10-inch skillet; cook over low heat, stirring constantly, until garlic foams and is sticky and straw-colored, 10 to 12 minutes. Off heat, add remaining tablespoon raw garlic, red pepper flakes, parsley, lemon juice, and 2 tablespoons pasta cooking water to skillet and stir well to keep garlic from clumping.
Transfer drained pasta to warm serving bowl; add remaining 3 tablespoons olive oil and remaining reserved pasta cooking water and toss to coat. Add garlic mixture and 3/4 teaspoon table salt or 1 teaspoon sea salt flakes to pasta; toss well to combine. Serve immediately, sprinkling individual bowls with portion of Parmesan, if desired.