  We study the fundamental bounds on precision measurements of parameters
contained in a time-dependent nonlinear optomechanical Hamiltonian, which
includes the nonlinear light-matter coupling, a mechanical displacement term,
and a single-mode mechanical squeezing term. By using a recently developed
method to solve the dynamics of this system, we derive a general expression for
the quantum Fisher information and demonstrate its applicability through three
concrete examples: estimation of the strength of a nonlinear light-matter
coupling, the strength of a time-modulated mechanical displacement, and a
single-mode mechanical squeezing parameter, all of which are modulated at
resonance. Our results can be used to compute the sensitivity of a nonlinear
optomechanical system to a number of external and internal effects, such as
forces acting on the system or modulations of the light--matter coupling.
