This paper examines the potential for reducing root-mean-square surface errors in shallow faceted reflectors by replacing flat facets with laterally curved membrane facets. Exact solutions are obtained for the small lateral deflections of equilateral triangular and rectangular membranes subject to isotropic tension and parabolic edge deflections. These solutions are used to minimize the rms error between a facet of a shallow paraboloidal surface and its approximating membrane facet. The resulting optimum placements and edge curvatures yield membrane facets which have significantly lower rms errors than the corresponding best-fit flat facets. The rms error reductions are about 55\% for equilateral triangles and 25-93\% for rectangles, depending on aspect ratio. The results suggest that replacement of flat facets with membrane facets conforming to curved structural members could yield reflectors with lower rms-error, or comparable error with larger facets and, hence, fewer structural members.