Can any plane in $\mathbb{R}^3$ be described using a two vectors that only move on two axes? Let's say I have the plane $2x-y-z=11$, and when I want the parametric form I get $u_1=(1,2,0)$ & $u_2=(0,-2,1)$. Aren't all vectors spanning the same plane parallel to one of these vectors? I can't make sense of this because that would mean that there would be no vectors on the plane that move on all three axes. The issue I have is that it seems I can always get two vectors that only move on two axes, no matter the plane, but obviously not all planes are spanned by vectors that only move on two axes.

Considering the example you have provided, assuming the standard basis $\left\{\mathbb{\hat{i}}, \mathbb{\hat{j}}, \mathbb{\hat{k}}\right\}$, the two vectors $u_1$ and $u_2$ have between them non-zero contributions in all components necessary to span the space. 
In this manner, the plane cannot be contained solely within any one of the $xy$, $xz$ or $yz-$planes. 
Hope this helps (and that I understood your question correctly). 
