Cook pasta in a large pot of boiling water until just tender or according to package directions. Reserve 1 1/2 cup of the cooking liquid and drain the pasta in a colander.
Meanwhile, finely grate 1 tbsp. zest and squeeze 1/4 cup from the 2 lemons; set the juice aside. Pat leek slices dry. Heat oil in a large nonstick skillet over medium-high heat. Add the leek, the lemon zest, 1/4 cup parsley, garlic, salt and pepper. Cook, stirring frequently, until the leek is lightly browned and softened, about 6 minutes.
Add the pasta, 1 cup of the reserved cooking liquid, the reserved lemon juice and the remaining 1/4 cup parsley to the pan. Cook, stiring constantly, until the liquid is mostly absorbed, 30 seconds to 1 minute. Add the remaining 1/2 cup liquid, if desired. Remove from the heat. Discard the garlic. Toss the pasta with 1/2 cup Parmesan and 2 tbsp. chives and serve with lemon wedges, if desired.