Position racks in the upper and lower thirds of the oven; preheat to 475 degrees F. Quarter the tomatoes, then scoop the seeds and pulp into a sieve set over a bowl (reserve the tomatoes). Press on the seeds and pulp to extract the juice (you should have about 1/2 cup). Chop the tomatoes and toss with 3 tablespoons olive oil on a rimmed baking sheet; season with salt. Toss the zucchini with 3 tablespoons olive oil on a separate baking sheet; season with salt and pepper. Roast, switching the position of the pans halfway through, until tender and browned in spots, 15 minutes.
Meanwhile, bring a large pot of salted water to a boil. Cook the pasta as the label directs; reserve 1 cup cooking water, then drain.
Heat the remaining 2 tablespoons olive oil in a large skillet over medium heat. Add the garlic; cook 2 minutes. Add the cherry peppers; cook 1 minute. Stir in the tomato paste, then the reserved tomato juice and 1/2 cup cooking water. Bring to a boil; simmer until reduced by half, 5 minutes. Add the pasta, vegetables, basil and parmesan to the skillet; season with salt. Cook, tossing, 1 minute, adding more of the reserved cooking water as needed to loosen.