Parking at the Grant Street Transportation Center in Pittsburgh (directly across the street from the Amtrak station) is enclosed, so you won't have to worry about digging your car out of any snow we might have over the weekend of your trip. I have parked there on a number of occasions and didn't have any particular safety concerns about the facility.
The train you are taking originates in Pittsburgh, so boarding there is a more leisurely process than it is in Greensburg.
i agree with the above post. i've parked in the garage across the street many times without worry. it's not free, but worth the few dollars you'll spend rather than trying to find a spot downtown or driving to greensburg.
i've never been to the greensburg station, but have never had any problems with either leaving from the pittsburgh station, picking people up, arriving, etc.
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