Let $V=\mathbb{Z}_p\times \mathbb{Z}_p$, where $p$ is prime. Let $\pi:V\rightarrow V$ be a map such that if two lines are parallel in $V$, then their images remain parallel.
This is a property of affine transformations on $V$, so my question is whether it is sufficient, i.e. given that $\pi$ satisfies the property above are we guaranteed to have an invertible matrix $M\in \text{GL}(2,p)$ and $a\in V$ such that $$\pi(v)=Mv+a$$ for all $v\in V$?