Place the chicken breasts, wild rice, chicken broth, carrots, celery, onion, garlic, thyme, salt, and black pepper into the slow cooker.
Stir the ingredients gently to combine, then cover and cook on low for 6 to 7 hours until the chicken reaches 165 degrees and the rice is tender.
Remove the chicken breasts, shred them with two forks, and return the meat to the slow cooker.
Stir in the heavy cream until fully combined, then let the soup warm for 10 more minutes on low.
Ladle the soup into bowls and top each serving with chopped fresh parsley.