  We construct two pencils of bielliptic curves of genus three and genus five.
The first pencil is associated with a general abelian surface with a
polarization of type $(1,2)$. The second pencil is related to the first by an
unramified double cover, the Prym variety of which is canonically isomorphic to
the Jacobian of a very general curve of genus two. Our results are obtained by
analyzing suitable elliptic fibrations on the associated Kummer surfaces and
rational double covers among them.
