While soft robots offer advantages in adaptability and safe interaction, their modeling remains challenging. This paper presents a novel, data-driven approach for model order reduction of slender soft robots using autoencoder-parameterized strain within the Geometric Variable Strain (GVS) framework. We employ autoencoders (AEs) to learn low-dimensional strain parameterizations from data to construct reduced-order models (ROMs), preserving the Lagrangian structure of the system while significantly reducing the degrees of freedom. Our comparative analysis demonstrates that AE-based ROMs consistently outperform proper orthogonal decomposition (POD) approaches, achieving lower errors for equivalent degrees of freedom across multiple test cases. Additionally, we demonstrate that our proposed approach achieves computational speed-ups over the high-order models (HOMs) in all cases, and outperforms the POD-based ROM in scenarios where accuracy is matched. We highlight the intrinsic dimensionality discovery capabilities of autoencoders, revealing that HOM often operate in lower-dimensional nonlinear manifolds. Through both simulation and experimental validation on a cable-actuated soft manipulator, we demonstrate the effectiveness of our approach, achieving near-identical behavior with just a single degree of freedom. This structure-preserving method offers significant reductions in the system degrees of freedom and computational effort while maintaining physical model interpretability, offering a promising direction for soft robot modeling and control.