Is there a way to see the serial number of the parts of my MacBook Air?

There's no way to get that information that I know of. Even applications like TinkerTool System, which can poll the management records, only list serial #s for the computer and the enclosure (which are the same serial #). Logic boards only report a model number, CPUs a chipset ID and there are no records at all for screens, keyboards, onboard devices or anything else that are likely bulk procurements by Apple's production centres.

Yes: System Information User Guide for Mac - Apple Support

If your MacBook Air is quite recent, you'll find this information here: Self Service Repair - Apple Support.