What conditions are sufficient to prove a transformation in $\mathbb{R}^2$ is affine?

I currently have shown that the transformation is bijective, points and lines are preserved and also the following:

If $M$, $N$, $X$, $Y$ are points such that $MN \parallel XY$ and $MN$ is parallel to one of the coordinate axes, then the images $M'$, $N'$, $X'$, $Y'$ also satisfy $M'N' \parallel X'Y'$ and $M'N'$ is parallel to one of the coordinate axes.

Do I still need to show parallel-ness in general is preserved or not?