This topic has come up several times before and there is always a lot of confusion.
Pressing PARK electro-mechanically actuates the rear brake calipers, just like many other modern vehicles. Tesla has an additional feature that may or may not apply additional pressure when you press and hold PARK a second time. When I do this on my car, I can hear the caliper motors apply, but I’m unable to determine if substantial additional force is applied.
Also, there is some conflating of terms.
Emergency Brake is when you press and hold the PARK button while the vehicle is in motion. The brakes will release when you release the button.
Parking Brake is when you press the PARK button while stopped, which applies the same brake mechanisms as the Emergency Brake function, but it keeps the brakes applied until you take the car out of park mode. A second application of the Parking Brake purportedly applies more pressure for really steep hills, but language about that has changed over time, and I’ve never heard of a Tesla rolling away on a steep hill.
So, to answer the original question, do you need to do the extra Parking Brake application? Probably not, but if you’re on a particularly steep incline, it wouldn’t hurt.