Remote Center of Motion (RCM) pivots are single-degree-of-freedom (1-DOF) rotational mechanisms whose axis of rotation lies outside their physical structure. They are widely used in surgical manipulators and precision optomechanical systems. This paper introduces a new family of planar linkages generating an exact RCM and exploits their adaptability to extend the performance limits of flexure-based RCM pivots through parasitic motion compensation. Three planar flexure implementations are investigated. The first realizes an exactly constrained RCM pivot with zero second-order parasitic motion and nearly constant rotational stiffness. The second exploits kinematic constraint redundancy to suppress parasitic motion while increasing radial and axial stiffness without compromising angular stroke or restoring moment linearity, thereby offering a conceptual solution to a longstanding trade-off in flexure pivot design. The third enables a large-stroke pivot with a rotational range of ± 45 ∘ , no internal degrees of freedom, and minimized parasitic motion. For each configuration, analytical models predicting kinetostatic properties are developed and validated through finite element simulations. The results demonstrate that controlling parasitic motion provides a powerful framework for designing high-accuracy, high support stiffness, and large-stroke flexure pivots.