In a 6 quart slow cooker, combine onion, carrots, celery, garlic, wild rice, bay leaves, thyme, salt, and pepper. Top with chicken breast halves. Add the chicken broth.
Place the lid on the slow cooker and cook on low heat setting for 6 to 6 1/2 hours or on high-heat setting about 3 1/2 hours. Carefully remove chicken and shred with two forks. Return to the slow cooker and stir. Remove the bay leaves. Add the parsley and season with additional salt and pepper, to taste.