Temporal graphs capture time-stamped interactions in domains such as finance and social networks. Prior work formalized historical k-core queries and the PHC index; constructing it reduces to computing, for every vertex and start time, its core time–the earliest end time at which the vertex enters the k-core of the windowed snapshot. The existing method computes start times serially and propagates neighbor updates, introducing dependencies within each round of computation, which hinder parallelization and scalability. We revisit the task through a parallel lens and propose a vertex-centric algorithm that computes core times for multiple start times in parallel, then updates them incrementally for subsequent start times, preserving exactness while mitigating these dependencies. Experiments on real temporal graphs show consistent speedups over a single-threaded baseline and a naive lock-based parallelization.