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A thinning algorithm is a connectivity preserving process which is applied to erode an object layer by layer until only a “skeleton” is left. Generally, it is difficult to prove that a 3D parallel thinning algorithm preserves connectivity. Sufficient conditions which can simplify such proofs were proposed recently inCVGIP: Image Understanding(59, No. 3 (1994), 328–339). One of the purposes of this paper is to propose a connectivity preserving fully parallel 3D thinning algorithm. The other purpose is to show how to use the sufficient conditions to prove a 3D parallel thinning algorithm to be connectivity preserving. By this demonstration, a new generation of 3D parallel thinning algorithms can be designed and proved to preserve connectivity relatively easily.