Heat oven to 450 degrees. On rimmed baking sheet, toss zucchini, tomatoes, and garlic with 5 tablespoons oil; season with salt and pepper. Spread into single layer; roast until zucchini is lightly browned, 20 to 25 minutes.
In large pot of boiling salted water, cook pasta until al dente, about 12 minutes. Drain pasta, reserving 1/2 cup cooking water. Return pasta to pot. Toss with remaining tablespoon oil; cover.
Pour reserved pasta water onto baking sheet of cooked vegetables; scrape with a wooden spoon to loosen browned bits. Stir vegetables and juices into pasta. Add parsley and Parmesan; toss to combine. Serve with more Parmesan.