Quadrupedal Movement (QM) is an integral part of Parkour training, because it involves:
1) looking funny,
2) while being in an awkward position,
3) while in front of other people.
Oh, and it also improves strength, coordination and flexibility of the arms, legs and core all at once. So here's one of my current favorite QM moves: the Inchworm (and yes, it involves steps 1 to 3 above). The inchworm is actually as hard as you make it, so it's a great exercise to scale up or down.
The principle is very simple: start standing, reach down in front of you until your hands reach the ground, flat.Then, walk you hands without moving your feet, as far as you can. Then walk your feet in, keeping your hands in place. The hard part is to keep your legs straight, and of course to not touch the ground with anything but your hands and feet, both flat (no fingertips!). After that, it's all about how much you can spread out and fold back in, like an inchworm. Decide beforehand of a distance to cover, nothing too long at first (that's a pretty slow way to move, and body parts will start complaining pretty fast). The further you can spread out and fold in, the fewer moves you need to make to cover a given distance. But it's harder. Trade-offs, trade-offs.
I recommend warming up your shoulders before doing it as this requires a good range of motion (e.g. do a few shoulder circles or pull-ups). It's a great exercise for building up core strength, but you should figure that out pretty quickly!
And did I mention it looks funny?