In this paper, a visual graphic approach is presented for the mobility and singularity analysis of mechanisms with no helical pair. The presented method is established upon the reciprocal screw system theory. Using the visual graphic approach, the mobility and singularity analysis mainly requires applying a few simple rules and involves into no formula derivation. As a case study, the mobility and singularity analysis is implemented for a class of two degrees of freedom (DOF) rotational parallel mechanisms (RPMs), including the Omni-Wrist III with four limbs and its two derived architectures with three limbs called the T-type and Δ-type RPMs. The Δ-type one is found to has kinematic properties close to the Omni-Wrist III.