I think it applies to both the auto as well as the manual transmission. The reason being is that you do not want to put a lot of stress on the locking pawl or pin in the auto transmission.
When you strap the front tires down on the dolly, they are still free to shift slightly as you turn corners. Notice that no matter how well you secure the car to the dolly, you will need to tighten it again after just a few miles.
With the wheels shifting or rolling in relation to each other, the auto transmission parking lock can be placed in a bind to the point that you will have a hard time gettng it out of park or damage the parking lock.
Amateur Radio Operator (KE5DFR)|Full-Time! - 2012 6.7L Ford Crew Cab Dually -2013 HitchHiker Champagne 38RLRSB - Travel with one Standard Schnauzer and one small Timneh African Gray Parrot