Place the chicken breasts, wild rice, chicken broth, carrots, celery, onion, garlic, thyme, salt, and pepper into the slow cooker.
Stir once to distribute the ingredients evenly, then cover and cook on low for 6 to 7 hours.
Remove the chicken breasts, shred them with two forks, and return the meat to the slow cooker.
Stir in the heavy cream and let the soup warm for 10 more minutes on low.
Ladle into bowls and top each serving with fresh parsley.