Place the chicken breasts, wild rice, chicken broth, carrots, celery, onion, garlic, thyme, salt, and pepper into the slow cooker.
Stir gently to combine the ingredients, then cover and cook on low for 6 hours.
Remove the chicken breasts and shred them with two forks.
Return the shredded chicken to the slow cooker and stir in the heavy cream.
Cook uncovered on high for 15 minutes to let the soup thicken slightly.
Ladle into bowls and sprinkle with fresh parsley before serving.