Parole: What is this?
Parole is an interesting concept. Normally if a person presents themselves at a Port of Entry without the proper identity documents and requests asylum, the immigration officer may issue a I-94 card and parole the person into the United States so that they can seek asylum and the safety of the United States. Is this considered an inspection, yes. It is however not an admission, although the person is physically able to enter the United States. They are considered an arriving alien.
An arriving alien is only eligible to adjust status infront of the USCIS and not an immigration judge. The way the rules are written, this type of inspection and issuance of an I-94 only confers jurisdiction upon the USCIS to adjudicate an adjustment of status application. It is an interesting concept because many individuals may have been allowed in the United States to pursue asylum but later lost the asylum. During this process and after a final order of deportation, that person may have gotten married to a United States Citizen and established a family and or other favorable equities. Under this doctrine, the person can adjust status, although not a formal admission it is an inspection.