A new parallel algorithm for detecting corners of planar curves or shapes is proposed in this paper. This algorithm is based on the morphological residue and corner characteristics analysis. The relationship between curvature radii and structuring elements is investigated. The location of the corners is detected based on the morphological residue set and the curvature extrema support region analysis. Noise influence is suppressed through the smoothing property of the algorithm. The algorithm works simultaneously on curves and shapes of multiple objects. The approach is different from chain-code based corner detection algorithms which need floating point computation for a curvature. For multiple objects, traditional algorithms deal with each curve individually, therefore, for multiply connected shapes or curves with intersection, coding and curvature computation are difficult and costly. The proposed algorithm deals with a whole image as a single object, therefore the computation complexity is significantly reduced. Our experiment demonstrates that the algorithm is fast and effective to execute on an SIMD parallel computer. This paper also presents a new parallel filling algorithm: a boundary-constrained morphological method for filling closed curves into shapes for corner detection.