  We study parafermion chains with $\mathbb{Z}_k$ symmetry subject to a
periodic binary drive. We focus on the case $k=3$. We find that the chains
support different Floquet edge modes at nontrivial quasienergies, distinct from
those for the static system. We map out the corresponding phase diagram by a
combination of analytics and numerics, and provide the location of $2\pi/3$
modes in parameter space. We also show that the modes are robust to weak
disorder. While the previously studied $\mathbb{Z}_2$-invariant Majorana
systems posesses a transparent weakly interacting case where the existence of a
$\pi$-Majorana mode is manifest, our intrinsically strongly interacting
generalization demonstrates that the existence of such a limit is not
necessary.
