Place onion in the bottom of a 6-quart slow cooker. Place chicken on top and sprinkle with Italian dressing mix.Add chicken broth.
Cover and cook on LOW for 4 to hours or on HIGH for 2 to 3 hours.
With 15 minutes of cooking time left, Discard about 1/2 of the cooking liquid. (So the sauce will be thick instead of runny.)Add the cream cheese, basil pesto, crushed red pepper flakes, black pepper, and Parmesan. Cover and continue to cook for the remaining 15 minutes.
Use two forks to break the chicken down into bite-sized pieces. Stir the sauce until the cream cheese is mixed in.