Do these tubes have a series heater? Some use a single heater wire for both sections, showing a short glowing segment at the top.
I'm speculating that with parallel heaters, the more massive power section cathode heats up more slowly, so that the driver has already stabilized - but with series, they might well heat up at the same time, so when the driver plate drops from B+ to its quiescent vale, that jump goes right into the power stage grid.
Do you also have C4S loads? That would make the above transition happen more rapidly.