There are some dangers in relying an a legacy Memo which could be revoked (it is now over 10 years old, so that seems unlikely).
That said, the Cronin memo is very clear on the issue.
That seems pretty clear that the use of an EAD is not required.
The case when obtaining an H1B extension subsequent to an entry using AP is also fairly clear.
If the H1B extension is granted, the Parolee "status" is terminated and the beneficiary of the extension returns to H1B status.
Of course, after using AP, there is the option to depart the USA and return using a valid H1B visa to re-establish H1B status.
I don't know what else to say. Your lawyer is being very conservative. I can only suggest making them aware of the actual
Cronin Memo. Another
link here.