Add the chicken breasts, chopped onion, minced garlic, crushed tomatoes, chicken broth, Italian seasoning, salt, and black pepper to a 6-quart slow cooker. Stir gently to combine.
Cover and cook on LOW for 6-7 hours or on HIGH for 3-4 hours, until the chicken is tender and cooks through completely.
Remove the chicken breasts from the pot and shred them using two forks. Return the shredded chicken to the slow cooker.
Stir in the uncooked pasta. Cover and cook on HIGH for another 20-30 minutes, or until the pasta is tender (al dente).
Stir in the heavy cream and half of the Parmesan cheese. Let it heat through for 5 minutes.
Ladle the soup into bowls. Top generously with the remaining Parmesan cheese, shredded Mozzarella, and fresh parsley. Serve immediately with crusty bread or croutons.