Here's a good image of the battle: http://allempires.com/articles/gaugamela/gaugamela0.gif.

Alexander had deployed his phalanx in an oblique fashion: basically, the right side of his phalanx was stronger and was intended to smash through the enemy formation. The right side of the phalanx would advance and hit the enemy before Parmenion's side would, which is why they are deployed 'at an angle'. Meanwhile, Parmenion's job was to hold the left side at all costs, which is why his left side would stretch throughout the battle into a semi-circle: he needed to protect his line from being encircled at all costs. Ironically, it was not Parmenion's left side that was flanked but his right, as the right wing of the phalanx advanced successfully while the left encountered heavy resistance, creating a gap between the two forces.

This maneuver/deployment is called "refusing the flank." (i.e. refusing to engage a flank you've made weak, by concentrating your force at the other flank)

From descriptions of the way the battle developed, it looks like Alexander planned this as a "break the enemy line" battle (as vs "envelop the enemy," as in Cannae). So he concentrated his force at the right, pushed hard against the Persian left, and when a gap or weak point developed in the Persian line, he used his companion cavalry to punch through it and drive straight at Darius, whom he correctly calculated as the Persian army's "center of gravity." I.e. drive Darius off the field, and the Persian army would lose the will to fight, their cohesion, and collapse.

Of course, if you make one wing of your army strong, that means your other wing must be correspondingly weak. So how to keep the weak wing from collapsing before your strong wing can push back the enemy wing in front of it? By delaying the decision on that wing as long as possible. I.e., hold it back, "refuse" to engage it, to give your strong wing as much time to win on its side as you can, before your weak wing comes under pressure.

This tactic is considered to have been first used by the Theban general Epaminondas, at the battle of Leuctra. During the brief period of Theban hegemony that followed this battle, Phillip II of Macedon, Alexander's father, lived as a hostage at Thebes. And he appears to have received much of his military education there.

So Leuctra is thought of as the first example of this tactic (oblique advance + refused flank + breakthrough), with Gaugamela being one of the finest examples of it from the ancient world. In a similar vein, the battle of Marathon is sometimes thought of as the first example of "double envelopment," with Cannae being the finest ancient demonstration of the technique.