  We define parafermionic observables in various lattice loop models, including
examples where no Kramers-Wannier duality holds. For a particular rhombic
embedding of the lattice in the plane and a value of the parafermionic spin
these variables are discretely holomorphic (they satisfy a lattice version of
the Cauchy-Riemann equations) as long as the Boltzmann weights satisfy certain
linear constraints. In the cases considered, the weights then also satisfy the
critical Yang-Baxter equations, with the spectral parameter being related
linearly to the angle of the elementary rhombus.
