You are welcome to request a travel document and to travel, as long as you do not return to your country from which you claimed asylum. It takes about 90 days to get a refugee travel document.
The above is intended only as general information, and does not constitute legal advice. You must speak with an attorney to discuss your individual case.
You really should speak to your attorney, if you had one. Traveling outside the US, will not affect your asylum status, provided that you do not return to your homeland.
It is always recommended that you apply for Advance Parole well in advance of your trip. All applications are done on a case by case basis....although usually it's between 90-120 days after filing. You can always try to request Expedited processing, but having to travel for work does not seem to qualify. USCIS website has information about this.
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