Place the chicken breasts, wild rice, chicken broth, carrots, celery, onion, garlic, thyme, salt, and pepper into the slow cooker.
Stir once to spread the ingredients evenly, then cover and cook on low for 6 to 7 hours.
Remove the chicken breasts from the slow cooker and shred them with two forks.
Return the shredded chicken to the slow cooker and stir in the heavy cream.
Cover and cook for 20 more minutes on low until the soup is heated through.
Ladle into bowls and top with chopped fresh parsley before serving.