If I accidentally hit a car that was parked illegally but the owner didn't want to call the cops or take my insurance information, what am I legally obligated to do?

The illegal parking may or may not break the causation chain of your accident. Meaning, you are still liable for the damage in my view. You can either settle with the owner or let your insurance company handle it. It is his problem not taking your insurance info, obviously wanting to hide the illegal parking but you do not have to agree to that.