Place the diced chicken in the bottom of a 6-quart slow cooker. Add the onion, carrots, celery, garlic, zucchini, diced tomatoes, tomato sauce, kidney beans, cannellini beans, and chicken broth.
Sprinkle in the Italian seasoning, dried oregano, garlic powder, salt, and black pepper. Give everything a good stir to combine the ingredients around the chicken.
Cover and cook on LOW for 6–7 hours or HIGH for 3–4 hours, until the chicken is cooked through and the vegetables are tender.
About 30 minutes before serving, stir in the uncooked pasta. Cover and continue cooking on HIGH for about 20–30 minutes, or until the pasta is tender.
Add the baby spinach and grated Parmesan cheese. Cover and cook for another 5 minutes, just until the spinach wilts and the cheese blends into the broth.
Taste the soup and add more salt or pepper if needed. Ladle into bowls and sprinkle with extra Parmesan cheese before serving.