First, so long as he appears at ALL immigration court hearings he will not be arrested simply for not having a lawyer. You need to contact his criminal defense attorney regarding the criminal case - it could be the charges were dropped or it could be the case is ongoing. Also consult with an immigration attorney who is familiar with criminal consequences on immigration.
Your husband can petition for his parents once he is a US citizen. Until then, they can try for tourist visas. They may also apply for investment based visas. Costs for any of the above vary by attorney and visa.
It appears that you "waived" the clock the day you submitted the asylum application. That is why the clock shows as "0". Unfortunately, you need for 150 days to elapse before filing for work authorization. Unless you an get the clock restarted, you will have to wait until your application for asylum is approved before you will be given work authorization.