Preheat the oven to 425 degrees F (220 degrees C). Press 1 pastry crust into a 9-inch pie plate; refrigerate along with second pastry crust.

Combine chicken, carrots, green peas, and celery in a medium saucepan; cover with water. Boil until chicken is no longer pink in centers and vegetables are fork-tender, about 15 minutes. Remove from heat, drain, and set aside.

Melt butter in a separate medium saucepan over medium heat. Add onion; cook until soft and translucent, about 5 minutes. Stir in flour, salt, black pepper, and celery seeds; slowly stir in chicken broth and milk. Simmer over medium-low heat until thick, about 5 minutes. Stir in parsley. Remove from heat and set aside.

Transfer chicken mixture to crust in pie plate; pour broth mixture over top.

Cover pie with remaining pastry crust. Seal edges and cut away excess dough. Make several small slits on top to allow steam to escape.

Bake in the preheated oven until pie is golden brown and filling is bubbly, 30 to 35 minutes. Cool before serving for 10 minutes.