Where to park ?

There's a garage underneath Millennium Park with easy access to the Art Institute.

You could use spothero app to locate nearby garages

I would park somewhere else in the city and take the train down.

The Art Institute of Chicago website lists a few options nearby, such as: Millennium Park garage (enter at Columbus Drive and Monroe Street), with 8’ clearance in limited sections and an elevator at the exit Grant Park north garage (enter at Michigan Avenue between Madison and Randolph), with 8' 3" clearance in limited sections and an elevator at the exit