Chirality is an inherent property of the nature, referring to the asymmetric property of an object, which represents an effective means of achieving compactly arranged structures bestowed by biological evolution. It opens up a novel avenue for designing programmable helical structures capable of switching between rotational and translational movements by the exhibited mirror-symmetric characteristics. This paper focuses on the topological manifold based parametric design of chiral origami mechanisms, combining the excellent folding features and periodic tessellation of origami structures. The mechanisms exhibit mirror-symmetric characteristics, allowing the transition among two distinct fully-folded configurations and the unfolded configuration. The proposed parametric design theory for chiral origami mechanisms combines the characteristics of spherical manifolds without length-scale constraints, conducting the angle synthesis with tuning 2D planar profiles and 3D motion space using only one design parameter. The proposed theory reveals the mathematical essence of chiral origami mechanism design, elucidates the intrinsic connection between key angle parameters and comprehensive mechanism design, and formulates a unified mathematical model for a class of chiral origami mechanisms units. It lays the foundation for the subsequent development of reconfigurable modular chiral origami robots with diverse motion capabilities and remarkable ability to adapt to various tasks and changing environments.