Add the chicken breasts, marinara sauce, diced tomatoes, chicken broth, onion, garlic, and Italian seasoning to a large slow cooker (6-quart or larger). 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 slow cooker and place them on a cutting board. Use two forks to shred the meat, then return the shredded chicken to the pot.
Stir in the uncooked fusilli pasta. Cover and continue cooking on HIGH for 20-30 minutes, or until the pasta is tender (al dente).
Reduce heat to LOW. Stir in the heavy cream (if using) and the grated parmesan cheese until melted and incorporated.
Turn off the slow cooker. Sprinkle the shredded mozzarella cheese over the top and let it sit covered for 2-3 minutes until the cheese melts.
Ladle the soup into bowls. Top with a dollop of sour cream (or yogurt) and garnish with fresh parsley. Serve immediately with crusty bread.