Warm oil in a large saucepan over medium-high heat. Add carrot, onion and celery and cook, stirring often, until slightly softened, about 3 minutes. Add garlic and sauté 1 minute longer. Stir in tomatoes and potatoes. Pour in broth and bring to a boil. Reduce heat to low and simmer, uncovered, until flavors have blended and potatoes are slightly softened, about 10 minutes.

Add zucchini and squash and continue to cook until potatoes are cooked through and vegetables are tender but not mushy, about 7 minutes. Season with salt and pepper. Add Parmesan, thyme, and parsley and stir well. Serve hot. Garnish with greens onions. Salt and pepper to taste.