ambidexiterity and dual-wielding...
well... duelists are fencers, swashbucklers - anything along that vein, I suppose. I've never seen fencers tumbling in the middle of the fight, but it fits well with the derring-do antics of fantasy swashbucklers, so why not?
Dual-wielding makes more 'sense' for fencers than tumbling - though some people contest the whole 'swinging-from-chandeliers' part of heroic swashbuckling, the use of bucklers, parrying daggers, main-gauche, etc. fits well with the tradition of fencing, as far as I know.
Off-hand parry is not a requirement for the Duelist class, since not every foil, epee, or rapier wielder is going to use an off-hand parrying weapon... but it fits the concept very well

That's probably what they meant by, 'a duelist can defend, but not attack with, a weapon in her off hand while using Precise Strike'