Final answer:
The regulator is the part of the anesthesia equipment that engages the harpoon of the syringe to permit aspiration.
Step-by-step explanation:
The part of the anesthesia equipment that engages the harpoon of the syringe to permit aspiration is the regulator. The regulator controls the pressure of the gas or liquid being delivered to the syringe. When the regulator is engaged, it allows the harpoon of the syringe to be activated, pulling in the desired substance through aspiration.