During the scene where the mechanics of the box are explained, Abe mentions that whatever is in the box moves parabolically between 'time A' and 'time B'. What does he mean by this? I've been assuming that time moves linearly between the two time points, so if the time was plotted on a graph it would look like a triangle wave. However, now I'm thinking the passage of time should be described by a different function, is there some sort of parabolic sine wave function that could be used to model it?
There's an explanation here. As I understand it, it's the field that grows parabolically, not time itself.
The time travel boxes used in the movie work like this. When the box is switched on, a "channel" begins in spacetime. That is point A. Call it 12:00pm. Over the next few seconds or minutes (depending on the box size) an electromagnetic field builds up gradually in a parabolic form until it plateaus.
Later, the box is turned off. This causes the field to diminish parabolically to zero. This closes the channel at point B, let us say at 12:01pm.
As you note, their model has time bouncing back and forth between the "future" when something is put into the box after it powers down, and the "past" of when it powered on, which as you note, does operate a bit like a sine wave, with the only safe spot to "exit" being at the top or bottom of the sine function where the "velocity" is 0 and the field is off (there's also a growing chance of spontaneous ejection, which in the movie works out to about 1300 minutes as a maximum time).
When the box is turned on the field builds gradually. During this build-up there is a period during which the box is still turned on but the field inside it is still weak enough that an object can exit or enter it, and thereby drop into the indeterminate channel. This narrow window of a few minutes is point A. The channel persists until the box is switched off later. When this is done, the field diminishes and again there is a window of a few minutes in which an object can enter or exit the box safely. This is point B.
Operating the box becomes a matter of timing and preparation. The operator, let's say Aaron, turns on the box at the time he wishes to exit and then walks away so that he is not present for window A. Later, Aaron returns to the box and switches it off. As the machine powers down, window B opens, and he climbs inside. Time passes and the machine switches off entirely. At this point, instead of exiting, Aaron "bounces" off the far end of the channel and begins to loop around backwards in time instead of forwards. Because time is now running backwards, the box appears to power up again from Aaron's point of view. Aaron waits until the box appears to power down again (in reality, this is point A, where it powered up), and then exits the box through window A in a timely fashion. He is now Aaron-2.