The other option might to park near the shops at the final T-junction, where you take the left for going to Nandi base. I think it should be ok to park there. Due to the shops, there are folks there all the time, so chances of someone doing some mischief with the car should be low. Or talk to one of the shop guys & see what they say. Request them to keep an eye on the car (if you are parking it where they can see it from their shop).
Otherwise, you can park at the Kamat restaurant, on the highway, where you turn left to go towards Nandi Hills. You can pay the security chap a bit in advance, to make sure they are extra watchful. I think it's ~14kms from there to the Nandi base. It should serve as a nice warmup ride, before starting the Nandi climb. (& nice wind down, after the Nandi descent, though the return leg may feel a bit long.)
-{db}.