In almost all algorithms for Model Predictive Control (MPC), the most time-consuming step is to solve some form of Linear Quadratic (LQ) Optimal Control Problem (OCP) repeatedly. The commonly recognized best option for this is a Riccati recursion based solver, which has a time complexity of O(N(n3x+n2xnu+nxn2u+n3u)). In this paper, we propose a novel Brunovsky Riccati Recursion algorithm to solve LQ OCPs for Linear Time Invariant (LTI) systems. The algorithm transforms the system into Brunovsky form, formulates a new LQ cost (and constraints, if any) in Brunovsky coordinates, performs the Riccati recursion there, and converts the solution back. Due to the sparsity (block-diagonality and zero-one pattern per block) of Brunovsky form and the data parallelism introduced in the cost, constraints, and solution transformations, the time complexity of the new method is greatly reduced to O(n3x+N(n2xnu+nxn2u+n3u)) if N threads/cores are available for parallel computing.