I'd just note that Metra is the regional rail system, not a light rail system. Also there are two lines that run between Joliet and Chicago - the Heritage Corridor, which runs weekday rush hour only service on the same line as Amtrak and serves Chicago Union Station, and the Rock Island District, which runs 7 days/week on a different line than Amtrak and serves Chicago LaSalle St. Station.It's been a while but I have done just what you are planning many times.
There a parking lot intended for commuters south of the the station on the east side of the tracks. I think the cost is $2.00 a day.
BTW there are three ways to get into Chicago from the Joliet station.
#1 Amtrak has 5 trains a day that will take you into Chicago Union station located west of the loop.
#2 METRA (The regional light rail system) has 16 trains a day that will take you to the La Salle St station on the south side of the loop.
#3 The Joliet station is also served by multiple Bus services that can get you to Chicago.
Have a good time in ChiTown!
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