You do not need any permit to rockclimb in that area of the park. You can learn more about climbing permits: nps.gov/yose/planyourvisit/…. For all other rockclimbing information, including food storage and waste disposal: nps.gov/yose/planyourvisit/…. You can park your car along the road at the base of El Capitan, near the crossover road. However, you must follow all food storage regulations and can't leave any scented items behind. There are food storage lockers located near the El Capitan Bridge which is where you can park.
Best, Marea
National Park Service Yosemite National Park nps.gov/yose/
Be aware of the following: Camping at the base of the NWF of Half Dome or other backcountry walls is allowed with a valid permit. I'm not sure if you plan to camp anywhere at the base of your climbs but if you plan to you will need to get a wilderness permit (see the links I sent before). In terms of parking, you can park at the eastern most end of the Valley open to public traffic which is just past Curry Village at Trailhead Parking.
Best, Marea
National Park Service Yosemite National Park nps.gov/yose/
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