i don't have experience on parking ticket after car is towed.
But I think, they towed your car to a place that not allow to park. It is not right. It is there mistake. They should tow your car to a place that allow to park. They should pay for the ticket.
But I found it hard to phone them(either people towed your car, or the police). Because it always the machine answer your phone call.
i experience the same case as you did and finially I lose the case.At that time, you need to pay more. As I understood, for such case, the driver will have 99% chance to lose the case. The article says that it is the driver's responsibility to alert such snow clearing. You will have one chance to plea not guilty. But I guess you will lose.