Ensuring constraint satisfaction is a key requirement for safety-critical systems, which include most robotic platforms. For example, constraints can be used for modeling joint position/velocity/torque limits and collision avoidance. Constrained systems are often controlled using Model Predictive Control, because of its ability to naturally handle constraints, relying on numerical optimization. However, ensuring constraint satisfaction is challenging for nonlinear systems/constraints. A well-known tool to make controllers safe is the so-called control-invariant set (a.k.a. safe set). In our previous work, we have shown that safety can be improved by letting the safe-set constraint recede along the MPC horizon. In this paper, we push that idea further by exploiting parallel computation to improve safety. We solve several MPC problems at the same time, where each problem instantiates the safe-set constraint at a different time step along the horizon. Finally, the controller can select the best solution according to some user-defined criteria. We validated this idea through extensive simulations with a 3-joint robotic arm, showing that significant improvements can be achieved in terms of safety and performance, even using as little as 4 computational cores.