  We construct lattice parafermions - local products of order and disorder
operators - in nearest-neighbor Z(N) models on regular isotropic planar
lattices, and show that they are discretely holomorphic, that is they satisfy
discrete Cauchy-Riemann equations, precisely at the critical
Fateev-Zamolodchikov (FZ) integrable points. We generalize our analysis to
models with anisotropic interactions, showing that, as long as the lattice is
correctly embedded in the plane, such discretely holomorphic parafermions exist
for particular values of the couplings which we identify as the anisotropic FZ
points. These results extend to more general inhomogeneous lattice models as
long as the covering lattice admits a rhombic embedding in the plane.
