Autonomous vehicles need to estimate the relative poses, i.e., position and orientation, of the surrounding vehicles on the road with at least 50 Hz rate and cm-level accuracy for platooning and collision avoidance applications. The LIDAR/camera solutions currently used for vehicle pose estimation do not satisfy these rate and accuracy requirements, necessitating complementary technologies. Vehicular visible light positioning (VLP) is a highly suitable complementary technology due to its high rate and high accuracy, exploiting the line-of-sight propagation feature of the visible light communication (VLC) signals from LED head/tail lights. However, existing vehicular VLP solutions impose restrictive requirements, e.g., high-bandwidth circuit, base station and VLC waveform constraints, and work for limited relative vehicle orientations, thus, cannot be extended for pose estimation. This paper proposes a VLP-based vehicle pose estimation (VLP-VPE) solution that eliminates these restrictive requirements by a novel VLC receiver design and a novel pose estimation algorithm. The VLC receiver, named QRX, is low-cost/size, and enables high-rate VLC and high-accuracy angle-of-arrival sensing, simultaneously, via the usage of a quadrant photodiode. The estimation algorithm first uses two of the designed QRXs to determine the positions of two head/tail light VLC transmitters on a neighbouring vehicle via triangulation, and then determines the 2D pose of the vehicle based on these two positions. Sensitivity analyses and simulations using traffic data from the Simulation of Urban Mobility (SUMO) demonstrate that the proposed solution performs pose estimation at cm-level accuracy and kHz rate under realistic road and channel conditions, demonstrating its eligibility for platooning and collision avoidance applications.