Combine salt, pepper, garlic powder, onion powder, paprika, parsley, mustard, and celery seed in a small bowl. Stir until evenly combined. Lay drumsticks on a clean work surface and rub all sides with the spice mixture.
Preheat an electric smoker to 250 degrees F (120 degrees C).
Lay drumsticks directly on the smoker racks and smoke for 2 1/2 hours.