Rivnuts from Homestead, PA on 7/1/2020 9:14:23 PM:
There is only limited "drop-off" parking at the Amtrak station in Pittsburgh. There are some nearby public parking garages in which to leave your car, albeit at a quite steep price for that duration.
In response to earlier inquiries about leaving their car for a week in downtown Pittsburgh some have stated that the nearby Hampton Inn will allow you to leave your car there for the week if you stayed there, say, the night before departing on your bike trip. You should confirm that with them if that approach is of interest to you. There are a couple of other hotels near the train station you could contact as to whether that have similar policies.
I have suggested to others that rather than trying to park downtown, you could consider staying/starting at the Waterfront Shopping area in Homestead PA which is about 8 miles from downtown and through which the GAP Trail passes. There are hotels, groceries, a Lowe's, etc. with copious amounts of parking and pretty safe. If you really wanted to see the origin of the GAP at Point State Park downtown, you could ride from Homestead back into town to do so.
Alternatively someone could drop you off downtown and then drive on out to the Waterfront, meet you there as you ride out of town and park the car. I would suggest parking at one of the hotels there or the Giant Eagle grocery or Lowes. I would think those would be the safest places to leave you car for the week.
Just my two cents worth. Others may have some good ideas as well.