If the key is not in the ignition and turned to "on", the airbag module is not awake and will not fire the airbags in the event of a collision while parked.
Another misunderstanding about your bags is why they don't go off in any type of accident. They only fire when needed, given the particular crash event. So if you get in a pure frontal accident, only your front airbags go off, and not the sides. Same is true of a side impact, only sides go off and not the fronts. If it's angular you might get both. And if the accident is not severe enough, you might not get bags at all.
Regarding black boxes it is a two way street. The OEM's put that functionality in the airbag module, and it can protect them or hurt them in lawsuits. That's because the airbag system records the entire vehicle history at the time of the accident... including faults / issues with any system in the car that talks on the communication bus. So for example if you pulled the fuse on your ABS or ESP system, that data is known and recorded in the module if that fault is present at the time of airbag deployment.
In today's world of lawsuits, any time there is a vehicle accident with a fatality, the families of the parties involved are often interested in this black box data to ensure the safety systems were working properly at the time of the accident.