Soft robots are challenging to control due to their nonlinear and time-varying dynamics. Data-enabled predictive control (DeePC) offers a model-free alternative by directly leveraging measured input-output trajectories to construct a predictive controller. However, its receding-horizon formulation requires solving a constrained optimization problem at every sampling instant, which can be computationally demanding for real-time deployment on resource-limited robotic platforms. To address this limitation, we propose an adaptive reinforcement-learning-based event-triggered DeePC (RL-ET-DeePC) framework for soft robotic control. A model-free RL policy is trained to determine when to invoke the DeePC optimizer based on the current system state representation, thereby reducing unnecessary optimization calls while preserving closed-loop performance. Simulation results show that RL-ET-DeePC reduces optimization frequency by up to 66% compared to periodic DeePC, while maintaining comparable tracking accuracy. Hardware experiments on a three-dimensional cable-driven soft robotic arm demonstrate zero-shot transfer, achieving a 34% reduction in optimization frequency with tracking accuracy comparable to periodic DeePC and more consistent performance than a static threshold-based event-triggered baseline.