There's a covered parking garage adjacent to the theater. If you get your parking ticket stamped at the theater or show your ticket to the checkout booth, parking is free. I THINK that's the way it works...it's been a while since I usually just use Lyft or Uber.
Tremont parking is free for events at the Paramount
I would suggest parking for free at a light rail station near you, take the train to Union Station and ride the free 16th mall shuttle up to the Paramount.
There are some public parking garages nearby that are not too expensive. 15th and glenarm is one. I recommend avoiding the surface lots as those tend to be more expensive.
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