In recent years, the field of soft robotics has received considerable attention due to its potential in increasing the safety of human-robot interaction. The design of soft robots possesses great challenges. For example, the longstanding challenge of co-design morphology and actuation makes designing them by hand a trial-and-error process. Earlier work presented by the authors proposes a computational design synthesis (CDS) method for the automated design of virtual, soft locomotion robot morphologies. This work extends the CDS method for morphologies with the automated co-design of actuation. Two methods are considered. In the first method, the actuation of designs is described by parametric actuation curves (PACs) that model feedforward actuation patterns. For every morphology in the design process, a set of PACs is optimized that assumes symmetric and cyclic gaits. The second method, soft actor-critic (SAC) reinforcement learning, removes this assumption as well as models feedback control for comparison. Adding PAC optimization to the CDS method is shown to improve the performance of the resulting designs and to achieve better results within less design iterations. SAC is, however, deemed less effective, due to the need for design specific problem tuning for each new morphology. The SAC experiments also show that the best found soft robot gaits are symmetric and cyclic, although this is not a constraint in the SAC problem formulation, thus verifying the assumptions made in the PAC formulation. To validate the search space modeled in the co-design CDS method, a state-of-the-art soft robot is replicated and compared.