Yes... sort of
Anyone booked in F class gets free access to the Board Room.... BUT you must note that just because you are flying first doesn't mean you are booked in F class. Complimentary & day-of upgrades (U class) are not included. Thats for sure.
As for A class, I believe that is a First Class mileage award. I don't think they can use the board room either. I don't know about after your flight.