The starter system in my car is dead

Automatic cars, by law, must have what's called a starter interlock which prevents them from being started in any gear other than Neutral or Park. This is a safety measure to ensure the car can't be started in gear and take off without the driver being in full control.

Which means your problem sounds like a coincidence rather than the result of trying to start in Reverse. What you can try is to start the engine in both Park and Neutral, rather than just Park. If it won't start in either of those positions, you may have either a problem with the starter system or the small switch that controls the interlock function.

But it's hard to imagine that tyring to start in Reverse would have damaged this interlock mechanism. Again, it sounds awfully like a coincidence. If it is the interlock system, you could try to remove the centre console to gain access to the gearshift mechanism and maybe spray some contact-cleaner onto the interlock switch, cycle it a few times and see if that restores power to the starter.