I'm still training at an A/G airfield which has regular parachuting.
I have always been taught to request engine start, turn off radios and then start, obviously turning radios back on after start.
It seems to be a good option to me due to parachutists and spinning blades not getting along particularly well
Whether this is particularly sensible at busier, ATC controlled fields is currently well outside my experience and I'd welcome guidance either way.