Bring at least 6 cups of salted water to a boil in a large pot. Cook for about 7-8 minutes or until al dente. Add the chopped broccoli during the last 1-2 minutes of cooking. Reserve about a half cup of the pasta water, then drain in a colander.
While pasta is cooking, heat the oil in a large skillet over medium heat. Add the onion and saute for 3 minutes. Add garlic and cook another minute.
Stir in the squash and corn and cook for another 3-5 minutes.
Add the cooked pasta and broccoli, pesto, and salt and pepper. Stir until coated. If it's too dry, stir in some of the reserved pasta water,
Add the tomatoes and heat through. Sprinkle each serving with parmesan cheese. Garnish with fresh basil leaves if desired.