The parachute system for the BMD and ASU-85 was triple canopy and used upward firing rockets to reduce the shock of impact. Additionally, the BMD had a chassis that allowed it to be lowered so that again the shock was supposed to be reduced. I actually still have "BMD air dropped" in my google history, so I think that if pictures exist they're sparse on the internet.
From what I gather the Soviets actually did exercises and did just what you talk about, got into their vehicles on the ground and drove off. I would fathom that on the BMD and ASU-85 that there were not sensitive systems to be damaged.
The M551 is another story
