Planetary boundary layer (PBL) parameterization schemes rely on empirical parameters that are typically assumed to be spatially and temporally invariant. In this study, instead of seeking a single optimal parameter as in conventional calibration, we investigate whether optimal PBL parameters exhibit flow dependence under controlled conditions. We consider the simplest prototype of flow dependency in the atmospheric PBL, specifically surface heating impacting the near-surface potential temperature profile. We apply a generalized stochastic inverse modeling technique to characterize the flow-dependent behavior of parameters governing vertical mixing and entrainment in a first-order turbulence closure, the YSU boundary layer scheme. Using the Weather Research and Forecasting (WRF) model in single-column configuration, we characterize model uncertainty from ensembles of perturbed simulations and observational uncertainty from LES output, treated here as the virtual truth providing synthetic observations. These two sources of uncertainty, together with prior knowledge on the parameters, are combined within the inverse estimation framework to obtain posterior distributions of PBL parameters. The results demonstrate that optimal PBL parameters vary with the atmospheric state. Multi-parameter experiments further reveal that different (independent) parameters can have correlated impacts on the modelled state of the PBL. The inferred joint posterior distributions provide a physically informed basis for representing parameter uncertainty in stochastic and perturbed-parameter schemes. Such dependencies are not captured by traditional perturbed-parameter ensembles that independently sample each parameter from a predetermined prior distribution. Beyond its application to uncertainty representation, we show that the stochastic inversion approach provides the theoretically optimal reference for evaluating ensemble-based parameter estimation methods. These methods become necessary for high-dimensional parameter spaces where the stochastic inversion becomes computationally intractable.