Numerical models of the ocean and ice sheets are crucial for understanding and simulating the impact of greenhouse gasses on the global climate. Ocean models rely on subgrid-scale parameterizations that require calibration and often significantly affect model skill. Comprehensive model sensitivities to parameters can be computed using approaches such as automatic differentiation. These can be used for calibration using gradient-based minimization of the misfit between model and data. However, differentiating legacy codes such as the Model for Prediction across Scales Ocean (MPAS-O) model is challenging. Based on the Simulating Ocean Mesoscale Activity (SOMA) configuration of MPAS-O used to represent an idealized, eddying, midlatitude, double-gyre system, we have created neural network-based surrogates for estimating the sensitivity of the ocean model to model parameters. We first generated perturbed parameter ensemble data for an idealized ocean model and trained three surrogate neural network models. The neural surrogates accurately predicted the one-step forward ocean dynamics, of which we then computed the parametric sensitivity. Future work extends this study by using adjoint-based sensitivity information obtained from a separate modeling framework, with a focus on elucidating parametric Atlantic Meridional Overturning Circulation (AMOC) sensitivities.