[en] The critical surface for the random-cluster model with cluster-weight $q⩾<!--\; ???\; -->4$ on isoradial graphs is identified using parafermionic observables. Correlations are also shown to decay exponentially fast in the subcritical regime. While this result is restricted to random-cluster models with $q⩾<!--\; ???\; -->4,$ it extends the recent theorem of (Beffara and Duminil-Copin 2012 Probl. Theory Relat. Fields 153 511–42) to a large class of planar graphs. In particular, the anisotropic random-cluster model on the square lattice is shown to be critical if $\frac{p_{\mathrm{v}}{p}_{\mathrm{h}}}{(1-<!--\; ???\; -->p_{\mathrm{v}})(1-<!--\; ???\; -->p_{\mathrm{h}})}=q,$ where p _v and p _h denote the horizontal and vertical edge-weights respectively. We also mention consequences for Potts models. (paper)