Suburban stations generally fall under the jurisdiction of their respective suburb, whereas stations in the city proper are administered by Metra (actually their parking subcontractor, which right now I believe is CPS parking).
My own personal experience is that the lots in the city are not maintained or administered very well by Metra's subcontractor. (In my opinion,) your best bet would be to find a suburban lot and call their village hall to confirm that they do have overnight parking, and advise you on how to pay for it. Another option is the outlying Amtrak stations - I know that multi-day parking can be purchased from the Amtrak agent at Homewood, and suspect that similar arrangements can be made at Naperville, Glenview, and Joliet.
That old car might be worth money!