Using a vegetable peeler, remove the zest from 2 lemons in long, wide strips; try to remove only the colored portion of the peel, not the bitter white pith beneath. Grate the zest from the remaining 2 lemons; reserve in a small bowl. Using a paring knife, cut about ½ inch off the top and bottom of one of the lemons and stand it on a cut end. Working from top to bottom, cut away the pith following the contours of the fruit, exposing the flesh. Now cut along both sides of the membranes separating the sections to free the segments. Remove and discard the seeds from the segments, then add the segments to a small bowl along with the juices. Repeat with a second lemon. Reserve the remaining 2 lemons for another use.

To the grated zest, add ½ teaspoon of the sugar and ½ teaspoon salt. Mix together with your fingers, then stir in 1 tablespoon of the oil; set aside. To the lemon segments and juice, add ½ teaspoon of the remaining sugar and ½ teaspoon salt; stir to combine and set aside.

In a large pot, combine 2 quarts water, 2 teaspoons salt, the remaining 1 teaspoon sugar and the zest strips. Bring to a boil, cook for 2 minutes, then remove and reserve the zest. Add the pasta and cook until al dente, about 3 minutes. Reserve 1½ cups of the cooking water, then drain the pasta and return it to the pot. Finely chop the zest strips; set aside.

In a 12-inch skillet over medium, cook the remaining 2 tablespoons oil, the garlic and pepper flakes, stirring, until the garlic is light golden, 1 to 2 minutes. Add the lemon segments with juices and cook, stirring, until fragrant and warmed through, about 30 seconds. Immediately add the mixture to the pasta in the pot along with ½ cup of the reserved pasta water, then toss to combine. Add the grated zest mixture, the parsley and the chopped zest, then toss again, adding more pasta water as needed so the pasta is silky and lightly sauced. Taste and season with salt and black pepper. Serve drizzled with additional oil and sprinkled with Parmesan.