All the new carparks don't have dedicated bike parking regardless of what the website says (Maybe there might be in the cars park H extension and the website might be reflecting this).
I work at the airport and have never seen a bike go through the boom gates. They all go in and out of the pedestrian gate at the northern end of the carpark. Another favourite trick for a lot of people is to tailgate you out the boom gates.
honestly, if you were going to be away for more then a week, it is cheaper to catch a taxi (even cheaper if you go Uber

), then it is to pay Perth Airport parking.