  We apply Witten's conjugation argument [Nucl. Phys. B 202, 253 (1982)] to
spin chains, where it allows us to derive frustration-free systems and their
exact ground states from known results. We particularly focus on
$\mathbb{Z}_p$-symmetric models, with the Kitaev and Peschel--Emery line of the
axial next-nearest neighbour Ising (ANNNI) chain being the simplest examples.
The approach allows us to treat two $\mathbb{Z}_3$-invariant frustration-free
parafermion chains, recently derived by Iemini et al. [Phys. Rev. Lett. 118,
170402 (2017)] and Mahyaeh and Ardonne [Phys. Rev. B 98, 245104 (2018)],
respectively, in a unified framework. We derive several other frustration-free
models and their exact ground states, including $\mathbb{Z}_4$- and
$\mathbb{Z}_6$-symmetric generalisations of the frustration-free ANNNI chain.
