I live a few miles outside of one of the JT national park entrances and make that climb 2-3 times a week. It starts to get hot in late May, but nothing too bad. Im going to go against the grain and recommend that if you decide to come up here that you ride into the park and not drive into it. JT has become an absolute zoo of humanity the last couple of years and designated parking spots can be hard to find at times, especially during the spring when it's still not too hot.
It's a beautiful ride though and a nice climb.