My tip is to go to Shinnick's Pub at 3758 S Union, buy a drink and ask politely for a parking pass for the street and they will give you one.
$3-$4 for a beer.
A six block walk to the park.
Shinnick's is a quiet neighborhood place, and you are not going to get blender drinks or craft beers. So don't ask them for one (or you will get "The Look")! It is mostly canned beer and bottles of Heineken and your basic mixed drinks (Jack & Coke, Vodka Tonic etc.)
Been going there for 20+ years, the only times I have had an issue is with getting a pass is Opening Day and, years ago, the Sox/ Cubs games when they would run out of street passes.
"Call your sons! Call your daughters! Call your friends! Call your neighbors! Mark Buehrle has a perfect game going into the ninth!"