Let me get this straight:
A. You came in with a F1 Student visa.
B. You applied for asylum after one year
C. You now work full time, I suppose based on work permit from B above.
Assumption 1: You are not going to school, or at least I assume that's the case. If that is is the case, your F1 student status is now invalid, you will have to rely on your asylum process to enter the US if you leave. On the other hand, if you are still going to school, and your university can still supply you with an I-20, verifying your school attendance, then there is no need for you to apply for advance parole...if your F1 visa hasn't expired. You can still use your F1 visa to enter the USA. If your F1 has expired, you can't leave the US since you will be required to go back to your country to apply for the F1 visa again, I'm sure you don't want to do that since that would invalidate your asylum application. Caution: F1 visa is a non-immgrant visa. When you apply for this visa, it is your intention that you don't intend to settle permanently in the USA. Even if this visa is valid, I don't know what the CP officers will think when they see that you applied for asylum.
Assumption 2: You have abandoned your F1 status completely. In that case, you need to apply for advance parole. Asylees with pending applications are required to apply for that I think. I don't know if you are eligible, but you would have to check the requirements.
It is very tricky to mix immigrant visas with non-immigrant visas when trying to enter the US. Its a different matter if its about staying here.
Good luck!