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Parametric quantum resonators provide a powerful platform where quantum phase
transitions arise from two-photon generation processes. These systems can
be driven from a low-excitation regime into a highly populated,
symmetry-broken phase, where nonlinearities play a crucial
role in determining the stable states. In this talk, we consider a
one-dimensional periodic chain of parametric resonators in two distinct
regimes - near and far beyond criticality - revealing two novel emergent
phenomena: a collective enhancement in critical quantum sensing and the
emergence of nonlinear topology. Near the critical point, we show that the
chain functions as a highly responsive quantum sensor, where
nearest-neighbor interactions induce a quadratic scaling of the quantum
Fisher information with the system size. This collective enhancement surpasses
the performance of an equivalent array of independent sensors. Beyond
criticality, in the absence of quadratic couplings, we show how nonlinear
nearest-neighbor cross-Kerr interactions may drive the system into a
spontaneously symmetry-broken topological phase, giving rise to topological edge
modes under open boundary conditions. The topology is dictated by the
structure of the Kerr nonlinearity and yields a non-trivial bulk-boundary
correspondence.