I'm far from an expert, but this is not Pletcher's first rodeo. Part of his success with Lawyer Ron was attributed to the use of draw reins:

http://cs.bloodhorse.com/blogs/horse-racing-steve-haskin/archive/2009/07/20/lawyer-ron-beyond-a-reasonable-doubt.aspx

Also, I don't think the use of draw reins is that uncommon. In the lead-up to either the Oaks or the Black-Eyed Susan, Zoe Cadman of XBTV mentioned that one of the fillies was difficult to gallop, necessitating the use of draw reins. When Always Dreaming was switched to draw reins, one of the reasons given for switching exercise riders was that Nick Bush was experienced with draw reins.