Hi Mike,
Interestingly, the eyes may also be playing apart, which may also be partially relevant to when you are in a car.
I think there is a mind set in both marching and driving activities that calls upon set spacial awareness so that you do not collide and others do not bump into you, too upset the focused journey.
The brain is heightened to clocking where you are as well as where you are going, and in marching the hands are perhaps like cat whiskers; ready to record physical objects that may suddenly get in the way, while at the same time defining clear space in a regimented manner.
Somewhere on this site was a link to a Ted talk, which showed a video of the change when a patient with PD got upon a bicycle and left all his symptoms behind.
Again, this is probably heightened awareness of space at work, along with the physical aspect in the position of the guys pelvis, which again points to this lower back area and activity around the Coccyx being very much involved in a common aspect related in PD.
Regards
Andy