DASH1, waiting for the stairs probably meant your 767 didn't get an aerobridge, hence a stand-off bay(quite unusual unless it was terminating or a shortage of a/bridges) where a delay on the ground may have been inevitable with a limited amount of stairs available in your situation-but then as you say,'Perth again'.
Whether or not an JQ A320 a/c is using QF stairs/ground equipment is nothing more QF's obligation fulfilling the ramp handling/service contract. If it was another contractor then that contractor would have to provide the appropriate ground equipment.