We use $L^2$-Higgs cohomology to determine the Hodge numbers of the parabolic cohomology $H^1(\overline{S}, j_∗\mathbb V)$, where the local system $\mathbb V$ arises from the third primitive cohomology of family of Calabi-Yau threefolds over a curve $\overline{S}$. The method gives a way to predict the presence of algebraic 2-cycles in the total space of the family and is applied to some examples.