I need some clarification on what being parallel to the xy, yz, and zx planes would mean when you are talking about parametric and symmetric equations.
The line $\displaystyle \ell(t):P+\vec{D}t$ is parallel to the $\displaystyle xy\text{-plane}$ provided $\displaystyle \vec{D}\cdot \vec{k}=0$.
The plane $\displaystyle (z-c)=0$ is parallel to the $\displaystyle xy\text{-plane}$.
There are similar statement for the other coordinate planes.