As noted in the other thread, weather on the Pass gets real interesting, real fast. I was up there today, went as far as the mile marker 52 drift (this one:
) and decided I didn't trust my side-hilling ability quite enough given the penalty for getting it wrong, plus I was only trying to ski Rock Creek Headwall (above the third switchback), and skinning the last half mile seemed a much safer choice.
I've seen it go from blue skies to 30 MPH wind and snowing in July in a matter of hours, and if you're on top and in the wrong place, you might not see the clouds approaching until they're within 15 minutes of hitting.
I've also sat out a thunderstorm in a snow cave because I misjudged how much time I had before it actually hit...I think that was July, or maybe August.
To your original question, the turn-out at the gate on this side is small enough that I'd be reluctant to park there overnight, even with a relatively small rig; if it gets snowed/blown in, the plow may be challenged to turn around, let alone clear the rest of the space. You could park down on Main Fork easily enough, and the snow coverage is decent now (road is sufficiently covered from the gate up to the third switchback, at least, although there's bare pavement in a few places and other spots where things are far deeper).
However, it's only two hours by road from Red Lodge to the Pilot Peak parking lot, and having your vehicle below you on the same side of the pass seems like a much safer plan than riding across unless—as noted—you know the terrain and nail the weather. The father of a coworker rode across from Red Lodge on Sunday (and back), said it took a couple of hours, but he's lived and played here for many years and knows the terrain.
FWIW, the snow up there right now is completely wind-f***ed, at least as far as I got today, and the top layer is super-reactive (I got propagating cracks in the 1/2"-ish windslab layer just trying to isolate a column to do an ECT). I'd be very wary of anything that's been loaded, even the lee side of the aforementioned MM 52 drift—if you slide down that way, you're either going to hang up on the guardrail or you're going to get down to the first switchback pretty darn quick, and there's a lot of big rocks still poking through down there).
Given the visible evidence of wind transport at the ridgeline, I'd assume it's even more wind-f***ed up top, but I don't know for sure.