Place chicken in the bottom of a 6 quart slow cooker along with onion, carrots, celery, garlic, bay leaves, thyme, rosemary, salt, and pepper.
Pour in chicken broth and cover. Cook on LOW for 3 hours.
Whisk the flour and cream together until smooth. Add it along with the milk, mushrooms, and rice to the slow cooker. Cook for 1-2 more hours on HIGH (or 2-3 hours on low) or until rice is cooked through. (Cook uncovered on high for the last 30 minutes for a thicker soup.)
Shred the chicken, ladle into bowls, sprinkle with parsley, and serve with bread.