  Quasi-parametric amplification (QPA) is a nonlinear interaction in which the
idler wave is depleted through some loss mechanism. QPA plays an important role
in signal amplification in ultrafast photonics and quantum light generation.
The QPA process has a number of features characterized by the non-Hermitian
parity-time ($\mathcal{PT}$) symmetry. In this report, we explore new
interaction regimes and uncover multiple $\mathcal{PT}$-symmetry phase
transitions in such QPA process where transitions are particularly sensitive to
external parameters. In particular, we demonstrate the feasibility of detection
of $10^{-11}$ inhomogeneities of the doped absorber, which is order of
magnitude more sensitive than similar measurements performed in a linear
absorption regime. In doing so, we reveal a family of $\mathcal{PT}$-symmetry
phase transitions appearing in the QPA process and provide a novel nonlinear
optical sensing mechanism for precise optical measurements.
