Gray-Box (GB) models are being considered for risk monitoring of Small Modular Reactors (SMRs). Their effectiveness is linked to the proper selection of the model parameters. This paper proposes a systematic methodology for identifying the most influential parameters of a GB model for estimating safety-critical variables of an SMR during normal operation and accident scenarios. The GB integrates a reduced-order physics-based model (White-Box, WB) with a data-driven (Black-Box, BB) model that corrects the outputs of the WB using the condition-monitoring data collected by sensors positioned onto the SMR. The proposed method combines signal decomposition, specifically the Hilbert–Huang Transform (HHT), and global sensitivity analysis (SA), based on first-order Kucherenko indices, to quantify the contribution of non-stationary, correlated GB input parameters to the variability of the safety-critical output parameters of interest. The proposed approach is applied to the Small Modular Dual Fluid Reactor (SMDFR), and the obtained results demonstrate its effectiveness in identifying informative and physically interpretable features, reducing complexity and computational burden to enable real-time risk monitoring.