We address multiparameter quantum estimation for coherently driven nonlinear Kerr resonators in the presence of loss. In particular, we consider the realistic situation in which the parameters of interest are the loss rate and the nonlinear coupling, whereas the amplitude of the coherent driving is known and externally tunable. Our results show that this driven-dissipative model is asymptotically classical, i.e. the Uhlmann curvature vanishes, and the two parameters may be jointly estimated without any additional noise of quantum origin. We also find that the ultimate bound to precision, as quantified by the quantum Fisher information (QFI), increases with the interaction time and the driving amplitude for both parameters. Finally, we investigate the performance of quadrature detection, and show that for both parameters the Fisher information oscillates in time, repeatedly approaching the corresponding QFI.