You can bring a cooler of your own food/drink into the park and leave it in one of many unreserved picnic pavilions, saving your spot at a picnic table. There will be a whole bunch of other coolers there too and nobody messes with them. You can bring in beer and drink it wherever but not in glass bottles.
There's a public trail called the Lagoon trail that runs around the outside of the park, part of which used to be the tracks for one of Lagoon's train rides. It's kinda weird and it shows some views of the park that most people never see. The trail starts in the campground.
Oh, and the Utah Olympic park with its zip line, slope tubing, alpine slides, etc is a different place than the Park City Mountain Resort with its zip line, alpine coaster, alpine slides, etc. It seems like people were confusing the two maybe. There's another alpine coaster, zip line and alpine slide at Snowbird too if you're really into it.