Data-driven modeling of coupled aeroelastic systems is limited by data availability: experiments are expensive, and high-fidelity simulations are computationally costly. Accurate surrogate models are therefore essential for rapid aeroelastic analysis, design, and control. In low-data regimes, nonlinear system identification must leverage known dynamical information and active sampling techniques to remain tractable. We present a physics-informed, data-efficient framework for identifying nonlinear, parametric aeroelastic surrogates using Sparse Identification of Nonlinear Dynamical Systems (SINDy). Known nonlinear structural dynamics are extracted from governing equations, and accurate reduced-order models (ROMs) capturing dominant vibration modes and their nonlinear couplings isolate the structural nonlinearities. This allows the data-driven model to focus solely on the unsteady aerodynamic forces, the primary source of uncertainty in coupled aeroelastic systems. Isolating the aerodynamic discrepancy simplifies nonlinear identification and enables compact, stable surrogate models for analysis and control. Known dynamics are incorporated via physically consistent constraints during regression and parametric-dependent stability priors, ensuring learned surrogates respect known stability trends across operating conditions. Given the high cost of data, adaptive sampling across the parametric regime is critical. Training datasets are chosen to capture both nonlinear responses and stability evolution, enabling accurate, generalizable surrogate models from minimal experiments or simulations. By combining cost-aware sampling with physics-informed nonlinear modeling, this approach yields fast-to-train, interpretable surrogates suitable for aeroelastic analysis, prediction, and control in data-limited environments.