Q,
Parolee is its own peculiar situation. It can't be a status because, technically, the person never leaves the POE and is not admitted into the USA.
At least with AP, you actually get an I-94 when you use it to enter the USA.
AOS is not a status either.
It is a period of Authorized Stay granted by the Secretary of DHS while the I-485 is pending adjudication.
Therefore, technically, a person relying solely on AOS is in unlawful status, but at the same time does not accrue any illegal presence. Everything is tolled while the I-485 is pending adjudication by virtue of USCIS policy.
The concepts are very difficult to understand, but are explained in this
USCIS memo.
I warn anyone who might read it, that it is not easy reading and is likely to give you a headache.