Flexure pivots, which are widely used for precision mechanisms, generally have the drawback of presenting parasitic shifts accompanying their rotation. The known solutions for canceling these undesirable parasitic translations usually induce a loss in radial stiffness, a reduction of the angular stoke, and a nonlinear moment-angle characteristics. This article introduces a novel family of kinematic structures based on coupled n-RRR planar parallel mechanisms which presents exact zero parasitic shifts, while alleviating the drawbacks of some known pivoting structures. Based on this invention, three symmetrical architectures have been designed and implemented as flexure-based pivots. The performance of the newly introduced pivots has been compared via Finite Element models with that of two known planar flexure pivots having theoretically zero parasitic shift. The results show that the newly introduced flexure pivots are an order of magnitude radially stiffer than the considered pivots from the state of the art, while having zero parasitic shift properties and equivalent angular strokes. These advantages are key to applications such as mechanical time bases, surgical robotics, or optomechanical mechanisms. Polymer mockups and a titanium alloy prototype have been manufactured for future experimental validation.