Imagine the parallelogram with vertices $(1,0,0),(2,1,1),(0,2,2),(1,3,3)$. How do we find parametric equations for it?
I found the Cartesian equations of its four sides
$$y=z=2-2x$$ $$y=z=5-2x$$ $$y=z=x+2$$ $$y=z=x-1$$
How do I go from these to equations $x(u,v),y(u,v),z(u,v)$?
So, for a point internal to a (planar) parallelogram as yours $$ \mathbf{x} = u\,\mathbf{s}_\mathbf{1} + v\,\mathbf{s}_\mathbf{2} \quad \left| {\;0 \leqslant u,v \leqslant 1} \right. $$ where $\mathbf{s}_\mathbf{1}$ and $\mathbf{s}_\mathbf{2}$ are the vectors corresponding to two concurrent sides, and $\mathbf{x} $ the position vector vs. the common point