I got my asylum case receipt, Now, can I apply for parole travel document before fingerprint?

You can apply for permission to travel, however it is not guaranteed.

Yes, you can apply for a travel permit.

Just beware that a parole travel document is never a guarantee of re-entry, and you should not travel to your home country. Also, extended stay outside the U.S. in a third country may negatively affect you asylum application.

Your request to travel is at the whim of USCIS. Your requests to travel will be heavily scrutinized when they review your case, ESPECIALLY if you travel back to your home country or any country with the same conditions as your home country. Why do you need to travel? It's entirely unnecessary if you even care about your asylum application. You also NEED to hire an immigration attorney. ALL asylum seekers MUST hire an attorney. An attorney can do the following: 1. Review your application 2. Submit documents and declarations to help you meet the LEGAL STANDARD which I'm pretty sure you may not know, yet you may have applied anyways 3. Attend the interview with you 4. Make arguments for you after the interview that can make or break a close case. Do not hire a paralegal make sure it is a licensed attorney,