Nonlinear model predictive control (NMPC) has been widely adopted to manipulate bilinear systems with dynamics that include products of the inputs and the states. These systems are ubiquitous in chemical processes, mechanical systems, and quantum physics, to name a few. Running a bilinear model predictive control (MPC) controller in real time requires solving a nonconvex optimization problem within a limited sampling time. This article proposes a novel parallel proximal-point Lagrangian-based bilinear MPC solver via an interlacing horizon-splitting scheme. The resulting algorithm converts the nonconvex MPC control problem into a set of parallelizable small-scale multiparametric quadratic programming (mpQP) and an equality-constrained linear-quadratic regulator problem. As a result, the solutions of mpQPs can be precomputed offline to enable efficient online computation. The proposed algorithm is validated on a simulation of an HVac system control. It is deployed on a TI LaunchPad XL F28379D microcontroller to execute speed control on a field-controlled dc motor, where the MPC updates at 10 ms and solves the problem in 1.764 ms on average and at most 2.088 ms.