Where is the PART NUMBER on my MacBook Pro? all I have is a serial number

Apple usually goes by the computer SUB-MODEL:

because model numbers often span multiple years so are less precise.

Your model number is on the case bottom near the hinge but you will need a lens to read the Zen-sized font.

M1 2020 Macbook Pro:

Mid 2012 Macbook Pro non-Retina:

In the second image, the model A1278 was used for at least four years although the capabilities of the computers changed significantly during that time. Not a unique identifier like the sub-model.

Ventura works of these SUB-MODELS:

I say "bad choice" on Apple's part.

Those numbers reflect configuration variants—localization; keyboard; storage; colors; etc—and Apple seems to hide it on Mac computers. Oddly enough, it shows in iDevices under Settings > General.

The model identifier shows in "System Report" available from "About this Mac.." :

When I must look up those "part numbers" (they call them "order numbers" in other settings) I use the MacTracker database (free in the Mac App Store). It shows this for the same model:

As for "what Apple will do?" questions, you need to ask them. We are your fellow end users and can be just as perplexed as you. 😉

What you need is your model ID - in your example, it is Macbook Pro 15,4

That is what is needed and found easily. And it is the identifying info you need.

We are all other end users and do not have access to Apple's way of doing anything, if you wish to use the internally used obscure part numbers, they may or may not show in the invoice Apple sent you.