<p>Biological swimmers such as fish achieve efficient thrust by modulating the stiffness of their body and pectoral fins via tendons and muscles. However, dynamic stiffness control remains underexplored in robotic swimming. We present a programmable online stiffness modulation (POSM) mechanism for aquatic robots: a multi-link arm capable of performing time-varying joint-stiffness control. The mechanism design allows each joint to operate within a different stiffness range. To achieve accurate, stable, and differentiable simulation, we leverage discrete variational mechanics to model the multi-rigid-body dynamics with external hydrodynamics forces. We then leverage the differentiable dynamics model to perform trajectory optimization over the joint stiffnesses for various swimming gaits. We achieve thrust improvements over non-stiffness-tuning baselines by up to 89% for drag-based swimming and 19% for body-caudal fin (BCF) propulsion, which demonstrates the performance benefits of coordinated online stiffness variation across joints and over time for efficient bio-inspired swimming. We further validate these benefits with an untethered, fish-inspired robot that uses the POSM to perform both BCF and drag-based pectoral-fin swimming.</p>