Place the chicken breasts, wild rice, chicken broth, carrots, celery, onion, garlic, thyme, salt, and pepper into the slow cooker.
Cover and cook on low for 6 to 7 hours until the chicken is cooked through and the rice is tender.
Remove the chicken from the slow cooker and shred it with two forks.
Return the shredded chicken to the slow cooker and stir in the heavy cream.
Cook for an additional 15 minutes on low to warm the cream through.
Ladle into bowls and sprinkle with chopped parsley before serving.