Bring a large pot of water to a boil.
As the pasta water is heating, bring a small saucepan of water to a boil. Add the carrots and asparagus to the small saucepan and blanch for 2 minutes. Using oven mitts or pot holders, remove the small saucepan from the heat and pour the carrots and asparagus away from you into a colander set in the sink. Rinse under cold running water to refresh.
While cooking the vegetables, add the pasta and 1 tablespoon of the salt to the large pot of boiling water. Return to a boil and cook the pasta until al dente, stirring occasionally, about 12 to 15 minutes. Turn the stove off, and with a small ladle, transfer 1/4 cup of the cooking liquid to a small cup and set aside. Using oven mitts or pot holders, drain the rest of the pot away from you into a colander set in the sink. Place the pasta in a large bowl with the 1/4 cup cooking liquid and cover with plastic wrap or a clean kitchen towel to keep warm.
In a large skillet over medium-high heat, melt the butter and, when hot, add the olive oil.
Add the onion and cook, stirring, for 2 minutes. Add the garlic, and cook, stirring, for 30 seconds. Add the zucchini and squash, and cook, stirring, for 3 minutes. Add the cooked carrots and asparagus, and cook, stirring, for 1 minute. Add the tomato, peas, the remaining teaspoon of salt, and the pepper. Stir and remove from the heat.
Pour the vegetables over the pasta in the bowl. Drizzle with the extra-virgin olive oil, add the parsley, and toss to coat evenly. Sprinkle, if you like, with the Parmesan cheese. Serve immediately.