We present a new algorithm for solving linear-quadratic regulator (LQR) problems with linear equality constraints, also known as constrained LQR (CLQR) problems. Our method's sequential runtime is linear in the number of stages and constraints, and its parallel runtime is logarithmic in the number of stages. The main technical contribution of this paper is the derivation of parallelizable techniques for eliminating the linear equality constraints while preserving the standard positive (semi-)definiteness requirements of LQR problems.