  Critical quantum systems are a promising resource for quantum metrology
applications, due to the diverging susceptibility developed in proximity of
phase transitions. Here, we assess the metrological power of parametric Kerr
resonators undergoing driven-dissipative phase transitions. We fully
characterize the quantum Fisher information for frequency estimation, and the
Helstrom bound for frequency discrimination. By going beyond the asymptotic
regime, we show that the Heisenberg precision can be achieved with
experimentally reachable parameters. We design protocols that exploit the
critical behavior of nonlinear resonators to enhance the precision of quantum
magnetometers and the fidelity of superconducting qubit readout.
