Recent strides in model predictive control (MPC) underscore a dependence on numerical advancements to efficiently and accurately solve large-scale problems. Given the substantial number of variables characterizing typical whole-body optimal control (OC) problems —often numbering in the thousands— exploiting the sparse structure of the numerical problem becomes crucial to meet computational demands, typically in the range of a few milliseconds. A fundamental building block for computing Newton or Sequential Quadratic Programming (SQP) steps in direct optimal control methods involves addressing the linear-quadratic regulator (LQR) problem. This paper concentrates on equality-constrained problems featuring implicit system dynamics and dual regularization, a characteristic found in advanced interior-point or augmented Lagrangian solvers. Here, we introduce a parallel algorithm designed for solving an LQR problem with dual regularization. Leveraging a rewriting of the LQR recursion through block elimination, we first enhanced the efficiency of the serial algorithm, then subsequently generalized it to handle parametric problems. This extension enables us to split decision variables and solve multiple subproblems concurrently. Our algorithm is implemented in our nonlinear numerical optimal control library ALIGATOR. It showcases improved performance over previous serial formulations and we validate its efficacy by deploying it in the model predictive control of a real quadruped robot. This paper follows up from our prior work on augmented Lagrangian methods for numerical optimal control with implicit dynamics and constraints.