Cook the pasta in salted boiling water until just about tender. Drain, reserving ½ cup pasta cooking water.
Meanwhile, heat 3 tablespoons oil in a large skillet over medium heat. Add garlic and cook, swirling pan often, until just golden. Add red pepper flakes and continue cooking for 15 more seconds. Add wine and then clams. Increase the heat to high, cover and cook until clams open and release their juices, 3-6 minutes, depending on size of clams. As clams open, use tongs to transfer them to a bowl.
Add ¼ cup of the reserved pasta water to skillet; bring to a boil. Add pasta to pan. Cook over high heat, tossing constantly, until pasta is al dente and has soaked up some of the sauce from the pan. Add clams and any juices from bowl to pan, along with parsley, and toss to combine. (Add more pasta water if sauce seems dry.) Transfer pasta to warm bowls and drizzle with remaining oil.