For one thing two parallel vectors are co linear (on the same line), so if (a,b,c) and (x,y,z) are parallel there exists some coefficient K such that (a,b,c)=K(x,y,z). Moreover, if two vectors are parallel, their cross product is zero.
Having a vector start at the origin and go to some vector, <a, b, c>, and given a second vector in the same form <x, y, z>, is there a way to tell that they're parallel.
Graphing them out is the only way I can do this, I'm sure there's a better method for doing this task.