When designing the configuration of co-axial parallel robots, designers often overlook the impact of constraint singularities on robot performance. To investigate the constraint singularities that arise when R(SS)² subchains were applied to co-axis actuated parallel robot mechanisms. By employing the methodologies of Lie group theory and differential manifold theory, an in-depth analysis of the R(SS)² subchains was conducted, which informed their integration into actuation coaxial configurations, leading to the design of two novel types of co-axis actuated parallel robot mechanisms. Subsequently, a comprehensive analysis was performed on these robot mechanisms, covering their types of motion, constraint space, and constraint singularities. Comparative analysis with the Delta robot mechanism revealed general patterns regarding constraint singularities in co-axis actuated parallel robot mechanisms equipped with R(SS)² subchains. The findings indicate that the conditions for parallel robot mechanisms to form constraint singularities are mitigated when R(SS)² subchains are arranged in an actuation coaxial layout. The research results provide a crucial theoretical foundation for future studies on configuration design, trajectory planning, and singularity avoidance in co-axis actuated parallel robot mechanisms.