The message should be put in .resx file. Make the message to "The specified arguments should not contain null elements.". Also, please pass in the parameter name to ArgumentNullException.

nameof(args)

No exceptions in the file use Resx file.
Please confirm using Resx file in the case.

I think you need to differentiate an exception thrown due to user action from an exception that's thrown due to internal error only.
The InvalidOperationException thrown when _dirty == true can happen only when there is an internal error -- the parser gets reused somehow. It shouldn't happen and is not user facing.
The ArgumentNullException thrown here only because a user action -- the user pass in invalid args. It's user facing. For user-facing exceptions, the message should be localized eventually.

I see your point. The issue could be raised (1) only in host application, (2) only if the host application developer constructs args with null element. So it is not "end user action" but a bug in the host application that is "internal error" case too. It looks the same as if we passed a bug with _dirty and an user catches this edge case with non-localized message. Thoughts?

Anyone using a PowerShell public API is an end user. An internal error is an error that cannot be triggered by an end user. In this case, an end user cannot cause the argument parser to be reused.
I hope this makes it more clear to you.

Thanks for clarify!

Done.