The choice of parameterization in Nonlinear (NL) system models greatly affects the quality of the estimated model. Overly complex models can be impractical and hard to interpret, necessitating data-driven methods for simpler and more accurate representations. In this paper, we propose a data-driven approach to simplify a class of continuous-time NL system models using linear approximations around varying operating points. Specifically, for sparse additive NL models, our method identifies the number of NL subterms and their corresponding input spaces. Under small-signal operation, we approximate the unknown NL system as a trajectory-scheduled Linear Parameter-Varying (LPV) system, with LPV coefficients representing the gradient of the NL function and indicating input sensitivity. Using this sensitivity measure, we determine the NL system's structure through LPV model reduction by identifying non-zero LPV coefficients and selecting scheduling parameters. We introduce two sparse estimators within a vector-valued Reproducing Kernel Hilbert Space (RKHS) framework to estimate the LPV coefficients while preserving their structural relationships. The structure of the sparse additive NL model is then determined by detecting non-zero elements in the gradient vector (LPV coefficients) and the Hessian matrix (Jacobian of the LPV coefficients). We propose two computationally tractable RKHS-based estimators for this purpose. The sparsified Hessian matrix reveals the NL model's structure, with numerical simulations confirming the approach's effectiveness.