This discussion has been had multiple times on all the model sub-forums and for good reason - the manual is overly simplified, and does not clarify the physical difference between the two modes clearly, which blows the whole thing open to speculation.
Here's my understanding based on various teardowns I've seen on the brake assembly for the model Y and the model S from trusted sources like Munro and from other informed users on here:
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 automatically.
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 hydraulic brakes are doing if they're functional. This is primarily for emergency situations for exactly the same 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 brakes for whatever reason.
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. This is true of all current models, it's a standard safety feature - you can't not have a redundant system for braking.
The hydraulic brake system is the same as most other automatics, aside from the fact that the newer models use the Bosch I-booster, which allows AP/FSD to use the hydraulic brakes 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. It does not "apply extra pressure", because that's not a thing with electric parking brakes, they're on or off, and apply a standard force based on the potential loaded weight of the vehicle. In any case, with parking brakes, unless they're just not functioning at all, the grip level from the tyre will always be your first point of failure. It's not difficult to lock the rear wheels completely on modern brakes. It does give you peace of mind if parking on a hill however that the transition between the two braking modes which happens by default does not cause the car to roll unexpectedly (but again - the parking brake would catch the car eventually, but this could still cause someone a fright, or to slip).
So TLDR;
- Parking brakes are independent of the hydraulic brakes on ALL cars - for good reason
- Putting the parking brake on manually before you get out does not apply extra braking pressure - it still only applies to the rear wheels (watch any video of Tesla's sliding down frozen driveways - the fronts roll, the rears slide) but it does help ensure the transition between the hydraulic brakes and the parking brake doesn't cause issues on steep hills if you so choose
- There is no way of parking and leaving the vehicle with all four wheels braked - hence the safety advice that you should never park on a hill when traction is compromised
