In this work, we propose a differentiable programming approach to data-driven modeling of distribution systems for electromechanical transient stability analysis. Our approach combines the traditional ZIP load model with a deep neural network formulated as a constrained nonlinear least-squares problem. We will discuss the formulation, setup, and training of the proposed model as a differentiable program. Finally, we will compare and investigate the performance of this new load model and present the results on a medium-scale 350-bus transmission-distribution network.