The increasingly widespread use of parametric mathematical models to describe biological systems means that the ability to infer model parameters is of great importance. In this study, we consider parameter inferability in nonlinear ordinary differential equation models that undergo a bifurcation, focusing on a simple but generic biochemical reaction model. We systematically investigate the shape of the likelihood function for the model's parameters, analyzing the changes that occur as the model undergoes a Hopf bifurcation. We demonstrate that there exists an intrinsic link between inference and the parameters' impact on the modeled system's dynamical stability, which we hope will motivate further research in this area.