I’m piecing together what I’ve read on other threads here, personal experience with a 2017 S, and a tear down of the model s brake booster and master cylinder here but I believe the following is accurate (and makes sense for these vehicles):
Park - (single push, with vehicle stopped and foot on brake) disengages any drive and uses “hill hold” (basically keeps the motorised brake booster engaged - which is what AP/FSD uses to slow the car just like your foot would) so the car is stationery, and you don’t hear anything because the brakes are still engaged as they were when your foot was on the pedal. When you get out of the driver seat, the brake booster resets to its default position and the electric parking brake (rear wheels only, engages).
Emergency brake - (long push and hold) is intended for emergency stops, and would add the electric parking brake into the mix in addition to what the brake booster is doing.
Pushing and holding the emergency brake when in park is just activating the parking brake
before you leave the drivers seat. It is not adding extra braking force, because the brake booster will reset as per the default use case and the light in the dash will go out as usual. The light is just there to tell you why you are rapidly decelerating or hearing a horrendous noise if you push and hold the button by accident
To add some clarity for those who are confusing the two braking systems on these cars:
The electric parking brake is an entirely separate system to the standard hydraulic brake system on all four corners.
The hydraulic brake system is the same as most other automatics, aside from the fact that the newer model s cars use the Bosch I-booster, which allows AP/FSD to use the hydraulics with no vacuum available (or needed) for a traditional booster.
Applying the parking brake manually when parked does nothing but apply the parking brake before you leave the vehicle.
The reason Tesla added the ability to manually apply the parking brake (basically a handbrake) in emergency situations is for exactly the reason other automatic cars will let you apply the handbrake for emergency situations… it could save you a change of pants if you lose your hydraulic system for whatever reason.