Not sure about your Model S, but the manual for mine says the parking brake is engaged automatically when you put the car into Park. It says nothing that I can see about needing to press the brake pedal for three seconds:
"Parking Brake
"The parking brake automatically engages when you shift Model S into Park, and releases
when you shift into any other gear. Note: The parking brake operates on the rear
wheels only, and is independent of the pedal operated brake system.
"Warning: In snowy or icy conditions the rear wheels may not have sufficient
traction to prevent Model S from sliding down a slope, particularly if not using
winter tires. Avoid parking on hills in snowy or icy conditions. You are always
responsible for parking safely.
"Warning: Your Model S may display an alert if the road is too steep to safely park
on, or if the parking brakes are not properly engaged. These alerts are for
guidance purposes only and are not a substitute for the driver’s judgment of
safe parking conditions, including specific road or weather conditions. Do not
depend on these alerts to determine whether or not it is safe to park at any
location. ..."
The manual does mention the red parking brake symbol when you manually apply the parking brakes, and says that reversing that action (releasing the parking brakes) also puts the car into neutral. But I still see no mention of three seconds.