They recommended a Nike "Air Structure Triax" (that's what it says on the box anyway). They recommended it because as a beginning runner my form was horrible and the shoe has a very thick, wide, stable heel and I believe their thinking was that it would keep me from rolling an ankle.
It was horrible for parkour because of the thick sole and stiffness. Also, the tread was a rubberized waffle tread glued on to the foam sole, and that shredded right off after about two cat leaps. The foam underneath was soooo slippery I might as well have not even been using my legs in parkour at all.
The "recommended" shoe for parkour is different for everybody but in general you want a good fit, a flexible shoe, some cushioning but not too much, a sole that won't shred off (grip is nice but as long as the shoe doesn't slide off of everything there is no huge need for any kind of massively grippy special rubber). Lightness and breathability are good too. If the sole is all in one piece, rather than tread patterns glued on, that is best. Plastic anywhere on the sole is dangerous.
A lot of people wear trail running shoes; I have heard that Raceblades (can't remember who makes them) are popular as well as Nike Dart and Free. These are shoes I have heard people talk about a lot but I haven't had any personal experience with them so I couldn't tell you one way or the other.
It would be worth it to look at the Ariakes and do a little compare/contrast with running shoes that are out there, to give you a place to start.
I personally wear kayaking shoes for parkour, the Ahnu "Bodega" and I like them a lot. This was after trying the ill-fated Nikes, and a pair of 5.10 Savants which were way too thick, stiff, and heavy.
There have been quite a few ladies on here who have given shoe reviews, and in the Consumer section there have been a lot of people discussing shoes. I would encourage you to read through those to get an idea of what's out there and what people like, to help you as you help these running stores.
Even better, try as many shoes as you can, although this does take time and $$.
Hope that helps. Good luck!