  We show that the following classes of lattice polytopes have unimodular
covers, in dimension three: the class of parallelepipeds, the class of
centrally symmetric polytopes, and the class of Cayley sums $\text{Cay}(P,Q)$
where the normal fan of $Q$ refines that of $P$. This improves results of Beck
et al.~(2018) and Haase et al.~(2008) where the last two classes were shown to
be IDP.
