Parafermions are Zn generalisations of Majorana quasiparticles, with fractional non-Abelian statistics. They can be used to encode topological qudits and perform Clifford operations by their braiding. We study the simplest case of the Z3 parafermion chain and investigate the form of the non-topological gate that arises through direct short-range interaction of the parafermion edge modes. We show that such an interaction gives rise to a dynamical phase gate on the encoded ground space, with the strongest order of the interaction generating a non-Clifford gate which can be tuned to belong to even levels of the Clifford hierarchy. We also illustrate the accessibility of highly non-contextual states using this dynamical gate. Finally, we propose an experiment that simulates the braiding and dynamical evolutions of the Z3 topological states with Rydberg atom technology.

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