You can obtain permit at the ranger station. There are forms just outside the building, you keep the copy with you, which you are supposed to return after you finish your climb.
You should also sign in on the board in the background - climber's name, name of the route, dates when you are going in and out. Many of the Black Canyon climbs take several days. Don't forget to erase your name after you completed your climb. You don't want park rangers to come and search for you.