Outside of the loop proper, I haven't found parking to be all that terrible to find. In the loop, using SpotHero makes it much easier.
Or visit a friend in Wicker Park by car and try to find street parking somewhere near their place.
You must just be really lucky or really skilled at parking.
I don't have much experience with going to the zoo on Saturdays, but I have been to Lincoln Park during Chipotle Cultivate, and again, parking was a few blocks away on Clark.
Call it bullcrap if you like, that's been my experience.