  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 $\varepsilon$ and
phase $\phi$. The results reveal that with the increase in $\varepsilon$, the
system undergoes a quantum phase transition from the normal to the superradiant
phase, with the critical point unaffected by $\phi$. 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.
