Pretty much, if the cross product is to have any of the properties we'd like it to have, then we're forced to conclude that the cross product of parallel vectors must be zero.

Reason #1: The cross product is perpendicular to both of its inputs

Two vectors form a plane. Every plane has just one direction that's perpendicular to it. We want the cross product to point in that direction.

“But wait,” you may be saying, “if the two vectors are parallel, then they don't form a plane, they just form a line.” Yep, exactly. So now there's a bunch of different directions that are perpendicular, so that means that the direction of the cross product is suddenly not well-defined, yikes.

Math's favorite thing to do when something is not well-defined is to cop out and give a weird answer that's vacuously true. I'm supposed to point in a bunch of directions at once? Ha-ha, math says, I can't do that, so instead I'm going to point in no direction at all!!

Reason #2: The cross product is anticommutative (because of the right-hand rule)

The cross product is supposed to follow the right-hand rule, but that ends up meaning it has to be anticommutative. (Point your first finger along i and your second finger along j. Where's your thumb pointing? Now point your first finger along j and your second finger along i. Now where's your thumb pointing?)

Ok, so, we're forced to admit that u x v = -(v x u). Well, what if we replace u with v, to find out what we'd get if we crossed any vector with itself? Then we're in a funny situation: v x v = -(v x v). That is, whatever v x v is, it has to be the same as its own negative – it can't change if you reverse it. The only vector that doesn't change if you reverse it is 0.

Reason #3: The magnitude of the cross product is the area of a parallelogram

So far our reasons have been about direction. There's also a good reason that comes from thinking about magnitude.

Two vectors form a parallelogram. (Make a copy of the first vector and stick its tail on the tip of the second. Then do the vice-versa thing to the second vector. Voila, a parallelogram.)

Vectors need both a direction and a magnitude. I think a pretty reasonable thing to say is the magnitude of the cross product is the area of that parallelogram.

“But wait,” you may be saying, “if two vectors are parallel, then they don't form a parallelogram.” You're kinda right. Parallel vectors do form a parallelogram, but it's a degenerate parallelogram: a real boring parallelogram that looks like the result of a regular parallelogram being fed into a hydraulic press. Guess what's the area of this degenerate parallelogram? You're right it's zero yay