Tie the parsley sprigs and bay leaves together with a piece of kitchen twine. Set aside.
Cut turkey kielbasa into long, quarter lengths; boil to remove excess grease. Remove from water and cut into bite-sized pieces.
Heat the olive oil in a 6-quart (or larger) soup pot over high heat. Add the onions, carrots, celery, and the parsley bundle, and cook for 2 minutes, stirring frequently. Add the garlic and mushrooms, and cook for 3 minutes. Add the broth, salt, and pepper. Cover the pot and bring to a boil. Remove the cover, reduce the heat to medium-low, and simmer for 12 minutes.
Add the zucchini, yellow squash, spinach, pasta and kielbasa to the soup. Simmer for 10-15 minutes.
Remove the parsley bundle and serve the soup hot, garnished with the grated cheese if desired.