I believe the "long-press Park" state automatically unlocks all the doors for convenience (as a couple of others have suggested).
The redundant "Park" indications are admittedly confusing ...
With their electric motors and direct-gear drivetrains, Tesla cars are more like traditional cars with manual transmissions in low gear when they are parked. An Emergency Brake is also a Parking Brake for these vehicles.
In traditional vehicles with automatic transmissions, a "Park" condition involves engaging locking teeth to prevent rotation of the drivetrain (that's what makes the awful noise if you try to shift into Park while you're still moving). The driveshaft is locked in "Park", but the individual wheels can still rotate. If a rear-wheel drive car is in "Park" and is jacked up off the ground, you can turn one rear wheel and the other rear wheel will turn in the opposite direction due to the rear differential. A car in "Park" doesn't roll since both rear wheels are on the ground.
Emergency Brakes typically consist of a redundant system to activate both of the rear wheel brakes of a vehicle. They function to slow and stop the vehicle in case the primary brake system fails. It is also a safe practice to set the Emergency Brake when parking a vehicle, in case the "Park" transmission lock fails.
So ... I experimented a little with my new Model 3 (v2018.42.4) and it's very clear that the first push of the button on the end of the gear selector/shifter stalk selects Park gear ("P"), engaging the rear brake caliper pads as a parking brake. A second "long push" of the stalk button causes a distinct additional brake action that sounds like the rear brake caliper pads are "tightening" even further for the securest possible parking, and the International "Parking Brake" icon/symbol appears on the display in addition to the large "P" (the car remains in Park).
From the Model 3 Owner's Manual:
"Shift Model 3 into Park by pressing the button on the end of the gear selector. The parking brake automatically engages ..."
"You can also unlock the doors by pressing the Park button on the end of the gear selector a second time. Pressing this button once
engages the Park gear and pressing it again unlocks the doors." (This is not necessary if you have "Unlock on Park" mode enabled)
"Parking brake - Electrically-actuated sliding parking brake integrated into rear caliper"
"Push the shift lever up or down to the first position and hold it there for more than 1 second to shift into Neutral gear, which allows Model 3 to roll freely when you are not pressing the brake pedal."
The key insight is that the Tesla Model 3 does not have a traditional Parking Brake/Emergency Brake (hand brake or foot brake). The car's anti-lock brake system has redundant hydraulic circuits (like aircraft) and is designed to be ultra-reliable (necessary for Autopilot, self-driving, etc). In the highly-unlikely event of a failure of the redundant hydraulic brake system, the driver could (in theory) perform a "long press" of the gear selector button to activate the Parking Brake/Emergency Brake and stop the car (I haven't tested this and it's not explicitly described in the manual, but it's logical). There's also a touchscreen menu where the Parking Brake can be enabled.
The Model 3 Owner's Manual refers to Park as a single condition/state. Manually applying the Parking Brake is considered a separate state, where the rear brake caliper pads are clamped down (tighter than in the standard "Park" state). Since the Tesla Model 3 has no manual hand or foot brake, this corresponds to manually setting the Parking Brake/Emergency Brake on a traditional vehicle.
In summary:
P = Park (drive motor(s) off with rear parking brake applied enough to prevent vehicle from rolling)
((P)) = Parking Brake applied (rear parking brake fully applied when red, system failure when orange, off when not displayed)
Best to place car in Park and then apply Parking Brake when car is parked on an incline