  We prove that the moduli space of cubic fourfolds $\mathcal{C}$ contains a
divisor $\mathcal{C}_{42}$ whose general member has a unirational
parametrization of degree 13. This result follows from a thorough study of the
Hilbert scheme of rational scrolls and an explicit construction of examples. We
also show that $\mathcal{C}_{42}$ is uniruled.
