Congrats on the job tinstaafl. Did not know you were in the US now.
The background check is not especially cumbersome, it consists of someone faxing a sheet of paper that you signed and waiting a week to get it back. They also email the individual now with the permission to begin 121 training. Lots of regionals hire foreign citizens.. mine included.
Like it was said.. go to
www.flightinfo.com and search past threads on this topic. you will find a lot of good info.
As for the advance parole, that is such bullsiht! I had to do that also. (I'm Australian by birth).