In a large mixing bowl, combine the beef, pork, bread crumbs, Parmesan, parsley, garlic, egg, salt, and pepper. Gently mix together with your hands. Shape the mixture into 42 (1-inch) balls, placing them on a baking sheet after shaping each one.
Heat oil in a large pot over medium-high heat. Add half of the meatballs and cook until browned on all sides, turning occasionally during cooking, about 3 minutes. Transfer the meatballs to a plate lined with paper towels. Repeat with the remaining meatballs.
In the same pot over medium-high heat, add the olive oil, carrots, onion, celery, and garlic. Cook stirring frequently, until the vegetables are tender, about 6 minutes.
Add the broth, salt, and pepper and bring to a boil. Stir in the pasta and meatballs.
Reduce the heat to medium-low. Cover and cook stirring occasionally until the pasta is tender, about 8 minutes.
Stir in the spinach and cook until the spinach wilts, about 30 seconds. Serve topped with grated Parmesan. Soup can be refrigerated for up to 1 week and frozen for up to 6 months.
