In my professional opinion an Advance Parole does not avail you to be a FAFSA recipient as a Lawful Permanent Resident as the LPR has not been approved and, if I am not mistaken, only US Citizens and LPRs are eligible for FAFSA funds.
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There are different kinds of parole and, you don't appear to be a parolee.
Plus, if you are an adjustment applicant, you probably have someone that filed an I864 ... which should make you ineligible for any income based benefits.
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Be careful about using advanced parole though, as sometimes those with advanced parole are nonetheless inadmissible back into the United States for a variety of reasons.