Bring a large pot of lightly salted water to a boil. Add pasta and cook until al dente, 8 to 10 minutes; drain.
Brown the sausage, stirring to crumble, in a skillet over medium heat. Remove sausage using a slotted spoon, leaving drippings in the pan. Cook and stir the onion and garlic in the sausage drippings until the onions are softened. Return the sausage to the pan. Stir in the red bell pepper, zucchini, yellow squash, and mushrooms. Season with salt and pepper. Cook another 5 minutes. Add the tomatoes, tomato sauce, thyme, basil, and oregano; cook until heated through, about 5 minutes.
Combine the drained pasta with the sausage mixture in a large bowl. Top with Parmesan cheese to serve.