  In two-dimensional statistical models possessing a discretely holomorphic
parafermion, we introduce a modified discrete Cauchy-Riemann equation on the
boundary of the domain, and we show that the solution of this equation yields
integrable boundary Boltzmann weights. This approach is applied to (i) the
square-lattice O(n) loop model, where the exact locations of the special and
ordinary transitions are recovered, and (ii) the Fateev-Zamolodchikov $Z_N$
spin model, where a new rotation-invariant, integrable boundary condition is
discovered for generic $N$.
