Bring a large pot of salted water to a boil. Add the pasta and cook as the label directs. Reserve 1 cup cooking water, then drain.
Meanwhile, heat the olive oil in a large pot or Dutch oven over medium heat. Add the garlic and cook, stirring occasionally, until golden, about 3 minutes. Add the clams, wine, corn kernels and cobs, red pepper flakes and 1/2 teaspoon salt; cover, increase the heat to high and cook until the clams open, about 7 minutes. Discard the corn cobs and any unopened clams.
Add the butter, lemon zest and pasta to the pot with the clams; toss to coat, about 1 minute. Add some of the reserved pasta cooking water to loosen the sauce if the pasta seems dry. Stir in the parsley and season with salt and pepper. Serve drizzled with olive oil.