Curve-thinning is an iterative reduction to extract centerlines from binary/segmented objects. Parallel thinning algorithms are composed of parallel reductions, and the fully parallel approach uses the same parallel reduction in each thinning phase. This paper presents the first fully parallel 3D curve-thinning algorithm acting on the unconventional face-centered cubic (FCC) grid. Our algorithm combines a sufficient condition for topology-preserving parallel reductions acting on (18, 12) pictures of the FCC grid with curve-endpoint preservation, hence its topological correctness is guaranteed for all possible pictures. In order to reduce the count of unwanted side branches in the produced centerlines, a novel iteration-level endpoint re-checking process is proposed.