Basics: Can Torque handle Parkour-Type-Games?

#1 Of course it can. The larger question is whether a team can handle the development of a game of that magnitude. A good start would be the 3D Action Adventure Kit add-on for Torque 3D.

#2 It can certainly be done, it just requires a lot of time and effort. The 3DAAK has some good features, but if you want Mirror's Edge you'll still be coding most if it yourself.

I've spent most of my time with Torque getting parkour movement in and working - here's a blog on the subject. Which proves that it can be done, somewhat ;P.

#3 Ok, what language(s) would I have to know to code it?

#4 Torque (all flavours) is written in C++, which you'll need to deal with quite a bit to implement anything complex. The good thing about movement is you're playing with gameplay code, not low-level stuff like data structures and rendering. The engines are scripted in TorqueScript, a typeless C-like language which is pretty easy to pick up if you know any programming already.

#5 What follows is my opinion, based on years of seeing great projects getting started but never finished. Feel free to ignore it. :)

There needs to be more (good) parkour games, so I look forward to seeing what you can come up with. But, before jumping into making your parkour game, I would recommend starting with something simpler.

Go through the tutorials and create a basic FPS style game. When you finish that, make a simple 3D platformer.

They don't need to be amazing, polished games when you're finished. The first couple games that anybody does are always...well...poor. You're doing these games to learn. And you're doing games that lots of other people have done because it will be easier to get help while learning.

Once you have a couple of games under your belt, you will know the right questions to ask. When that time comes, find me online and I can point you at some really interesting information on AI movement and animation. :)

Enjoy the journey.