Combine chicken, carrots, peas, and celery in a saucepan. Cover with water. Boil until chicken is no longer pink in the middle and vegetables are fork tender, about 15 minutes. Remove from heat, drain, and set aside.
Cook and stir onions in butter in a saucepan over medium heat, until soft and translucent, about 5 minutes. Stir in flour, salt, pepper, and celery seed. Slowly stir in chicken broth and milk. Simmer over medium-low heat until thick, about 5 minutes. Stir in parsley and remove from heat. Set aside.
Place chicken mixture in bottom pie crust; pour hot liquid mixture over. Cover with top crust, seal edges, and cut away excess dough. Make several small slits in top to allow steam to escape.
Bake in preheated oven until pie is golden brown and filling is bubbly, 30 to 35 minutes. Cool for 10 minutes before serving.