<p>Data-driven discovery of governing equations, say, sparse identification of nonlinear dynamics (SINDy) has gained a wide attention, while response data produced by typical nonlinear oscillators induces a further challenge for the data-driven approach, as they are essentially involved with multiple time scales or slow/fast mixed dynamics, i.e., slowly-varying amplitude is modulated by fast-varying oscillation. In this contribution, a demodulation algorithm is introduced to extract slowly-varying amplitude (envelope signal) before it is fed to the sparse identification procedure, enabling SINDy to directly discover a parsimonious model for parametric slow dynamics, i.e., evolving at a large time scale and explicitly depending on physics parameters. This approach significantly improves training/regression efficiency and becomes less sensitive to error due to numerical differentiation. Through applications to typical parameter-dependent nonlinear oscillators, a detailed comparison study with both testing data and standard multi-scale analysis (perturbation method) demonstrates efficiency and validity of the current data-driven approach for predicting slow dynamics with parameter dependence.</p>