Kodachrome Basin State Park does not allow climbing.
No climbing in Kodachrome State Park! Boundaries are not always clearly marked or fenced, check with the rangers before climbing anywhere near their park.
Northern extent of the Grand Staircase-Escalante National Monument. This is a large area, little explored by climbers. At first glance, the rock appears to be mostly colorful choss but better stone exists in places. Kodachrome Basin State Park encompasses most of the best-looking features, including some of the spindliest, coolest towers in the desert. But, lucky for the fragile material they are made from, climbing is illegal in the state park. Grosvenor Arch, a few miles south and east, is an attractive and popular feature but provides only easy scrambling. But in one or two places, outside the state park, the choss somehow forms monstrous pinnacles. Not for the faint-hearted; the shaley stone here makes the Fisher Towers look like Lost Arrow Spire. More recently, a few formations with nice cracks and quality rock are being found. There's surely more waiting to be discovered.
Drive to Cannonville, fill up with gas and supplies, head south out of town into the vast wilderness of the GSENM. Most of the choss worth getting out of the car for is probably going to be near Kodachrome Basin State Park. Climbing is illegal in the state park so if you try climbing at all, be sure that you you are outside the park. Climbing is allowed within GSENM. There are parcels of private property that may not be very well posted. For the GSENM folks who manage this enormous monument, climbing is the least of their concerns, let's keep it that way, keep a low profile, do some homework to check the ownership of the land you want to climb on. And have fun!