Place the chicken breasts in the bottom of the slow cooker. Add the wild rice, chicken broth, carrots, celery, onion, garlic, and thyme.
Season with salt and pepper. Stir gently to combine the ingredients.
Cover and cook on low for 6 hours or until the chicken is fully cooked and the rice is tender.
Remove the chicken breasts and shred them with two forks. Return the shredded chicken to the slow cooker.
Stir in the heavy cream until the soup becomes creamy and well mixed.
Ladle into bowls and top with fresh parsley before serving.