In this paper, we present a generic topological and geometrical framework which allows to define and control several parallel algorithms for 2D digital curve approximation. The proposed technique is based on combinatorial map simplifications guided by geometrical criteria. We illustrate the genericity of the framework by defining three contour simplification methods: a polygonal approximation one based an area deviation computation; a digital straight segments reconstruction one which guaranties to obtain a loss-less representation; and a moment preserving simplification one which simplifies the contours while preserving geometrical moments of the image regions. Thanks to a complete experimental evaluation, we demonstrate that the proposed methods can be efficiently implemented in a multi-thread environment to simplify labeled image contours.