- cheap shoes: Parkour is going to beat up your shoes rather fast, especially at the beginning. It's no use spending too much on a pair of fancy shoes until you know how long you can make them last.
- no plastic arch: many running shoes have a plastic support under the arch of the foot; those will hook on rails and make you slide to your doom. Don't do it. Continuous soles are safer.
- lighter is better: a thinner and flatter sole will allow you to feel the ground better and improve your grip. Thinner shoes have also more flexibility, while heavy shoes weight your feet down.
Beyond that, it's all about personal preference. We like the Feiyue wushu martial arts shoes because they're about 15€, ultra-thin and grippy, but they also absorb no impact whatsoever and you tear through the sole in a few months. Some have been happy with the Adidas Samba, which are sturdier and have a nice flat sole. Twio-X recommends the Kalenji Ekiden 50 from Decathlon, also for their low, low price. The Fivefinger barefooting shoes are cool but a bit too technical and expensive to my taste. There are many trail running (but not hiking!) shoes that are light and sturdy (we had some great La Sportiva, but they stopped making them) but once again pricey. You can find specialized Parkour and freerunning shoes from K-Swiss, FiveTen and others, also fairly expensive.
And in the end, the best way (but most challenging, and not recommended without a long and careful preparation) to train is barefoot!!