A holder of the TPS status is allowed to apply for advance parole: a document authorizing the alien to travel abroad and return to the United States.
Section 212(d)(5) of the Immigration and Nationality Act (“INA”) grants the Attorney General authority to parole into the United States aliens seeking admission “on a case-by-case basis for urgent humanitarian reasons or significant public benefit.” 8 CFR ยง 212.5(f) authorizes USCIS to grant advance parole to aliens who will travel without a visa, by issuing “an appropriate document authorizing travel.”
The significance of the parole is that if a person entered without inspection, received a TPS or DACA status, traveled on parole and was in fact paroled back to the US, the person is no longer an EWI, and, can adjust status to one of the permanent resident.
Now, the tricky part is that not everybody who left the country with the advance parole will be able to return: the Immigration authorities does not have to let you in, even if you have an advance parole. The USCIS has an authority to issue the parole, but it is the CBP, the border patrol, who verifies that the person is still entitled to it.
The Arrabally/Yerrabelly decision held that a person with pending adjustment of status application may travel on advance parole and come back with the triggering the 3/10 years bar of unlawful presence, because the travel on advance parole is not a departure for immigration purposes. The same rationale is applicable to the holders of the TPS status: a person with a valid TPS status may travel outside the US and return back using the advance parole without triggering the bars of inadmissibility associated with unlawful presence. Matter of _____, St. Paul, MN (AAO September 17, 2012).