Thinning is an iterative object reduction to extract skeleton-like shape features from binary objects. Parallel 3D curve-thinning algorithms are composed of parallel reductions to produce centerlines. In subfield-based parallel thinning the digital space is partitioned into \(s\ge 2\) subfields which are alternatively activated, and only certain points in the active subfield can be deleted. This paper presents the first two subfield-based parallel 3D curve-thinning algorithms acting on the unconventional face-centered cubic (FCC) grid. The proposed algorithms use partitionings of the the FCC grid into four and eight subfields.