This paper presents a parallelizable suboptimal Model Predictive Control (MPC) design framework for structured linear systems with polytopic state and control constraints. The proposed real-time control policy addresses structured large-scale quadratic programming (QP) problems by deriving the control action by evaluating a finite set of piece-wise affine functions (PWA). These PWA functions are precomputed offline as explicit solutions to small-scale multiparametric QP problems that tailor this method for industrial- oriented or embedded implementation. Prioritizing computational efficiency over optimality, the proposed MPC controller ensures real-time feasibility within stringent time constraints. The key contributions include the derivation of a lower bound on the fixed number of algorithm iterations required to guarantee the closed- loop performance under assumptions and an open-source C-code library, ParExMPC, based on the proposed framework. Numerical simulations highlight the scalability of the method, accommodating systems with a high number of decision variables and extended control horizons-well beyond the capabilities of existing explicit MPC methods. Furthermore, the developed implementation of the proposed close-to-optimal control method demonstrates superior runtime performance compared to state-of-the-art implicit MPC solutions, which rely on online optimization.