Bring a large pot of salted water to a boil. Add the pasta and cook as the label directs. Reserve 1/2 cup cooking water, then drain.
Meanwhile, heat the olive oil in a separate large pot over medium heat. Add the garlic and cook, stirring, until pale golden, about 3 minutes. Add 2 tablespoons chiles and cook, stirring, until the oil is red, about 1 minute. Stir in the cherry tomatoes and 1 cup water and simmer, stirring occasionally, until thickened, about 20 minutes.
Add the pasta, reserved cooking water, parsley and remaining 1 tablespoon chiles to the sauce. Cook, tossing, until well coated, about 1 minute; season with salt. Divide among bowls and drizzle with olive oil.