This paper presents a novel methodology to model and optimize trajectories of a quadrupedal robot with spinal compliance to improve standing jump performance compared to quadrupeds with a rigid spine. We introduce an elastic model for a prismatic robotic spine that is actively preloaded and mechanically lock-enabled at initial and maximum length, and develop a constrained trajectory optimization method to co-optimize the elastic parameters and motion trajectories toward enhanced jumping distance. Results reveal that a less stiff spring is likely to facilitate jumping performance not as a direct propelling source but as a means to unleash more motor power for propelling by trading-off overall energy efficiency. We also visualize the impact of spring coefficients on the overall optimization routine from energetic perspectives to identify the suitable parameter region.