While you are in the United States in TPS or U4U status, or while your asylum claim is pending, you cannot leave the country. Or rather, you can leave, but there is a very high risk that you will not be allowed back into the USA and you will lose your status. To prevent this from happening, you need to obtain an Advance parole travel document before leaving the United States, thanks to which you will be able to leave the United States and return back without losing your status.
To obtain Advance parole, you must submit Form I-131 to USCIS and pay the filing fee. In this case, there is no exemption from paying the fee for anyone. On average, you have to wait 1-1.5 years for consideration of this form.
But in emergency cases, USCIS can issue an Emergency advance parole, for example, in the event of a serious illness or death of a close relative. Of course, such an exit permit will be issued much faster.
When can I request expedited processing of Form I-131, Application for Travel Document, to obtain an Emergency Advance Parole?
In cases where the reason for expedited processing is professional, academic or personal obligation.
You can request expedited processing regardless of whether you have a planned trip or an emergency trip.
You can also request expedited processing if you need to travel out of the United States for a pre-planned event and you filed Form I-131 on time in advance, but the standard processing time still prevents you from receiving your travel document by your planned departure date.
You can request expedited Form I-131 processing using your USCIS online account. You will also be required to upload evidence of why you need expedited processing.