Given its highly nonlinear and multi-physics-coupling nature, modeling the startup process of a turboshaft engine has always been a difficult task, which is further aggravated on a turboshaft engine in a hybrid-electric system. With sufficient data from bench experiments, this article establishes a data-driven numerical model to simulate the startup process of a hybrid-electric turboshaft engine. Neural networks are organized hierarchically to estimate three key parameters of the engine system: engine speed, leftover torque, and exhaust temperature. Specifically, on exhaust temperature, different neural networks were tried and compared. The networks were built, trained, and tested in sections, and then tested as a whole. Compared with the mechanism-based methods, the model proposed in this article has a simple structure and can be built automatically. According to the running tests, the proposed method can give accurate results at a fast speed using small hardware resources, showing great potential as a reference model on real-time control units.