Advance Parole is for non-resident foreign nationals. It allows a non-resident to travel back to the United States without applying for a visa. Since an advance parole document does not replace the passport, the non-resident with an Advance Parole still needs a valid foreign passport to enter the U.S.
Advance parole is most commonly used when someone has a pending: Form I-485, Application to Register Permanent Residence or to Adjust Status Form I-589, Application for Asylum and for Withholding of Removal
Re-entry Permit is for residents. While it is valid, a re-entry permit allows the resident to apply for admission to the U.S. without having to obtain a returning resident visa from a U.S. Embassy or Consulate.
Permanent or conditional residents should apply for a re-entry permit if they will be outside the U.S. for one year or more.