In this article, we study the planar cubic differential systems with invariant affine straight lines of total parallel multiplicity seven. We classify these system according to their geometric properties encoded in the configurations of invariant straight lines. We show that there are only 17 different topological phase portraits in the Poincaré disc associated to this family of cubic systems up to a reversal of the sense of their orbits, and we provide representatives of every class modulo an affine change of variables and rescaling of the time variable.