I didn't get that impression. I think the parents felt that the FBI should have done more for them and for Billy considering his use to the FBI over several years prior to his trip to Russia.
Assuming the FBI asked him not to go, why and how should they FBI put resources into finding out what he did and what happened to him? Whether the FBI sent him there or not, he decided on his own to stay beyond his plan, and then enter a war zone. Everyone who does some contracting for the FBI deserves world-wide babysitting service?