1. Preheat oven to 450°.
2. Bring 4 quarts water to a boil in a large Dutch oven. Add 1 tablespoon salt and spaghetti to boiling water; cook 10 minutes or until spaghetti is al dente. Drain spaghetti in a colander over a bowl, reserving 1/3 cup cooking water. Return spaghetti to pan; set aside, and keep warm.
3. While spaghetti cooks, combine tomatoes, 1 tablespoon olive oil, vinegar, 3/8 teaspoon salt, and pepper on a jelly-roll pan, tossing well to coat. Bake tomato mixture at 450° for 10 minutes or until tomatoes are soft and lightly charred in places.
4. Add tomatoes and any tomato juice to spaghetti in Dutch oven. Add 3 tablespoons reserved cooking water to jelly-roll pan, scraping pan to loosen browned bits; carefully pour water mixture and remaining 1 tablespoon oil into spaghetti mixture. Place Dutch oven over medium heat. Add remaining reserved cooking water, 1 tablespoon at a time, until spaghetti mixture is moist, tossing frequently. Stir in basil and parsley. Sprinkle with cheese. Serve immediately.