Minimum-Lap-Time Planning (MLTP) problems, which entail finding optimal trajectories for race cars on racetracks, have received significant attention in the recent literature. They are commonly addressed as optimal control problems (OCP) and are numerically discretized using direct collocation methods. Subsequently, they are solved as nonlinear programs (NLP). The conventional approach to solving MLTP problems is serial, whereby the resulting NLP is solved all at once. However, for problems characterized by a large number of variables, distributed optimization algorithms, such as the Alternating Direction Method of Multipliers (ADMM), may represent a viable option, especially when multi-core CPU architectures are available. This study presents a consensus-based ADMM approach tailored to solving MLTP problems through a distributed optimization algorithm. The algorithm partitions the problem into smaller sub-problems based on different sectors of a track, distributing them among multiple processors. ADMM is then used to ensure consensus among the distributed computational processes. The paper also outlines specific strategies leveraging domain knowledge to improve the convergence of the distributed algorithm.The ADMM approach is validated against the serial approach, and numerical results are presented for both single-lap and multi-lap scenarios. In both cases, the ADMM approach proves superior for problem dimensions of 70k+ variables compared to serial methods. In planning scenarios with complex vehicle models on long track horizons, i.e., for problems with 1M+ variables, the efficiency gain of the ADMM approach is substantial, and it becomes the only viable option to maintain computational times within acceptable limits.