YouTube app won't update on iPad 3
Your iPad has been released in March of 2012 and is now eight years
old. It is normal that Apps and Apple will drop software support at some point.

iOS updates and App updates are different.

iOS is the Operating systems from Apple for their mobile devices and the newest version usually supports devices up to ~5 years old.
App updates (in this case YouTube) are released from the developer (in this case Google). They may require a certain iOS version (which includes certain libraries and features that the App can use).

You are experiencing the second problem, where the App needs a newer version of iOS to run. However, your iPad is not being supported by Apple anymore.
That means in your case that there is nothing you can do about the App update.
What you could try though, is to open YouTube in your browser. Usually this works but I can only confirm this for newer devices.