Add the chicken breasts, wild rice, chicken broth, carrots, celery, onion, garlic, thyme, salt, pepper, and bay leaf to the slow cooker.
Stir the ingredients gently to distribute the seasonings.
Cover and cook on low for 6 to 7 hours, until the chicken reaches 165°F and the rice is tender.
Remove the chicken breasts and shred them with two forks, then return the meat to the slow cooker.
Stir in the heavy cream, cover again, and cook on low for 15 more minutes.
Remove the bay leaf, taste and adjust salt if needed, then ladle into bowls and top with the chopped parsley.