  In this work we numerically study critical phases in translation-invariant
$\mathbb{Z}_N$ parafermion chains with both nearest- and next-nearest-neighbor
hopping terms. The model can be mapped to a $\mathbb{Z}_N$ spin model with
nearest-neighbor couplings via a generalized Jordan-Wigner transformation and
translation invariance ensures that the spin model is always self-dual. We
first study the low-energy spectrum of chains with only nearest-neighbor
coupling, which are mapped onto standard self-dual $\mathbb{Z}_N$ clock models.
For $3\leq N\leq 6$ we match the numerical results to the known conformal field
theory(CFT) identification. We then analyze in detail the phase diagram of a
$N=3$ chain with both nearest and next-nearest neighbor hopping and six
critical phases with central charges being $4/5$, 1 or 2 are found. We find
continuous phase transitions between $c=1$ and $c=2$ phases, while the phase
transition between $c=4/5$ and $c=1$ is conjectured to be of
Kosterlitz-Thouless type.
