If my car is parked over fresh (unsalted) snow, will it damage my 2005 Hyundai’s undercarriage?
Not likely. It’s the salt that starts the rust.
Thank you.
It is driving on the salted roadways that may cause a propensity to rust. If you are getting scared look at the number of older Hyundais still on the road.
Unless your car was stored where it is, you have plenty of salt there regardless. As long as it is cold, the snow will do little other then clog brakes and suspension components which is a bigger problem if it thaws slightly and refreezes as ice. When it melts, theoretically it can “help” the salt brine find it’s way to welded areas which are most suseptable to rust. So, the snow in itself is not the problem, but it is “aiding and abetting” the salt already in the area when it melts. Liquid is a great carrier for salt to find entrance to rust areas.
I’m just taking a wild guess, but you don’t live in Buffalo. right ? 
Nope.
But I suppose if it makes you feel better you can keep it in your bedroom.
Seriously, what choice do you have?