We introduce a reduced-order modeling strategy for nonlinear hyperbolic parametric systems that combines Proper Orthogonal Decomposition (POD) with the Discrete Empirical Interpolation Method (DEIM) and a Principal Interval Decomposition (PID) in time with time-averaging. The PID framework partitions the simulation horizon into principal intervals and constructs time-averaged states that serve as compact, representative snapshots for the reduced basis generation. Nonlinear contributions are efficiently approximated through DEIM, which yields a projection-based reduced model with a substantially lower online cost. Particular attention is given to preserving the well-balanced structure of the full-order formulation, so that equilibrium solutions and their associated hyperbolic behavior are consistently maintained at reduced dimension. The resulting POD–DEIM–PID model provides a significant reduction in computational effort while retaining close agreement with high-fidelity solutions over a broad range of parameter values. Numerical experiments on representative nonlinear hyperbolic problems confirm the robustness and efficiency of the proposed reduced-order framework.