The points $z_1,z_2,z_3,z_4$  in the complex plane are the vertices of a parallelogram taken in order, if and only if:

Diagonals of a parallelogram are bisected each other at a point

i.e.,         $\frac{z_1+z_3}{2}=\frac{z_2+z_4}{2}$

$\Rightarrow$              $z_1+z_3=z_2+z_4$ .