Parametrically driven nonlinear resonators represent a building block for realizing fault-tolerant quantum computation and are useful for critical quantum sensing. From a fundamental viewpoint, the most intriguing feature of such a system is perhaps the critical phenomena, which can occur without interaction with any other quantum system. The non-analytic behaviors of its eigenspectrum have been substantially investigated, but those associated with the ground state wavefunction have largely remained unexplored. Using the quantum ground state geometric tensor as an indicator, we comprehensively establish a phase diagram involving the driving parameter ε and phase ϕ. The results reveal that with the increase in ε, the system undergoes a quantum phase transition from the normal to the symmetry-breaking phase, with the critical point unaffected by ϕ. Furthermore, the critical exponent and scaling dimension are obtained by an exact numerical method, which is consistent with previous works. Our numerical results show that the phase transition falls within the universality class of the quantum Rabi model. This work reveals that the quantum metric and Berry curvature display diverging behaviors across the quantum phase transition.