Not as simple as that.
For the purposes of United States law, the receiver or frame is legally the firearm,[3] and as such it is the controlled part. The definition of which assembly is the legal receiver varies from firearm to firearm, under US law. Generally, the law requires licensed manufacturers and importers to mark the designated receiver with a serial number, the manufacturer or importer, the model and caliber. In addition, makers of receivers are restricted by International Traffic in Arms Regulations. Thus, in the case of a firearm that has multiple receiver parts (such as the AR-15, which has an upper and a lower receiver), the legally controlled part is the one that is serialized. For the AR-15 rifle, the lower receiver assembly is legally considered the actual receiver[4] (although it is functionally a chassis that also houses the separate trigger group); while in the FN-FAL rifle, it is the upper assembly that is serialized and legally considered the receiver.
In the case of the glock the frame of the pistol is the serialized part and considered the firearm. In the case of an AR15 type rifle, the lower receiver that contains the trigger assembly is the serialized part considered the firearm. There are rifles where the upper receiver is considered by the manufacturer to be the serialized part allowing for swapping of multiple lower receivers to accommodate caliber changes (like the SCAR and ACR). In none of these cases is the actual trigger pack or trigger assembly considered the serialized part, nor is it necessarily the part that contains the trigger assembly that will be considered the serialized part, though that is the most common case.
Basically the law just states that the serial number must be engraved on the frame or receiver ( in the case of AR-type firearms the lower receivera). Any extra serial numbers are just extra. So the serial number on the glock slide or barrel doesn’t make it the controlled part.
What it comes down to is that if you don’t know enough to know what part of the particular firearm you are engraving is the controlled pard, then don’t do it.