Heat the oil in a large saucepan over medium heat. Add the onion, carrot, and celery and cook, stirring occasionally, until tender, 5 to 6 minutes.
Add the broth and 2 cups water and bring to a boil. Add the orzo and cook until tender, 8 to 10 minutes.
Stir in the meatballs and spinach. Serve with the Parmesan.