It's quite important to me for a few perhaps off-kilter reasons, but I didn't actually realise that until I noticed this thread and decided to have a look.
One of the big pulls for the game for me is PvP, another is dungeon building. Parkour is relative to both, because it gives us an idea of how movement and spatial interaction will be. If combat works seamlessly with parkour, dextrous combat will be absolutely incredible. If jumping and general acrobatics hold to the same logic and gravity system as that displayed in Silver Run, you could in theory set up combat arenas devoted entirely to aerial combat, with fighters making passes at each other through the air while navigating, changing direction/elevation and seeking advantage through the use of poles and other props. Fighters could use walls, roof decor (read: chandeliers), furniture... even the ground-elevation level itself to gain an advantage in general combat. If you can do a knee-slide under a table and combo into an upward push to throw it at your opponent, or combo into a spin-slash with a heavier weapon to break the legs in order to disrupt the footing of anyone on it, to give an example... the whole system would allow a degree of freedom, choice, and room to improvise in combat that has literally never been seen before in a game of this sort.
That's just looking at a couple of very basic mechanic examples that would not require finicky coding for specific circumstances, but simply setting up general rules and manouvres and letting physics take care of the rest. With those sorts of mechanics, combat takes on the dynamic, hectic pace that is characteristic of great combat motion picture experiences and makes the fighting feel real, or better.
That's just PvP. For dungeon construction, you have a whole arena of possibility ranging from mildly dramatic to purely ridiculous. Log/piston traps designed to cripple would have the secondary, hilarious effect of propelling someone at high speed through the air; death drops could be designed in such a way that a dextrous enough character can wall-run the way down to the ground to avoid damage, taking advantage of concealed footholds to slow their descent enough to not die. Players could use traps and moving dungeon parts as props to grab and jump from - a swinging axe over a narrow bridge might be a shortcut to the top of the wall it stops at, for example, or a means to gain the speed you need to leap the last of the distance to the next bridge across.
There's a lot of possibility if they take that route. Too much to delve into, really. Even if not, simple parkour enables a great degree of freedom in movement and gives us a rough idea of the level of gravity in the game, which in turn means people can start planning their builds and combat styles accordingly. We might see combat styles less like Dark Souls and more like stuff like RWBY if the combat pacing and spacing is factored in by the pledgers who're looking to create martial arts.