This letter proposes a parallelizable algorithm for linear-quadratic model predictive control (MPC) problems with state and input constraints. The algorithm itself is based on a parallel MPC scheme that has originally been designed for systems with input constraints. In this context, one contribution of this letter is the construction of time-varying yet separable constraint margins ensuring recursive feasibility and asymptotic stability of sub-optimal parallel MPC in a general setting, which also includes state constraints. Moreover, it is shown how to tradeoff online run-time guarantees versus the conservatism that is introduced by the tightened state constraints. The corresponding performance of the proposed method as well as the cost of the recursive feasibility guarantees is analyzed in the context of controlling a large-scale mechatronic system. This is illustrated by numerical experiments for a large-scale control system with more than 100 states and 60 control inputs leading to run-times in the millisecond range.