The A320 uses different engines (turbofans) which cannot be shut down yet. So no, not with the next update anyways. Shutting down turbofan engines is high up on the to-do list though, which means that, fingers crossed, the A320, B737, B747 and Learjet 45 will get their full engine simulation with shutdown and restart soon.
A lot of the systems that depend on the engines have already been added. That means as soon as the A320 for example gets the new engines that can be shut down it will be able to reach cold and dark status. And so does the Learjet.
I understand, thank you for the info!