This study presents an analytical framework for nonlinear dynamic modeling and free vibration analysis of a double parallelogram flexure mechanism. The dynamic model is developed using the beam constraint model, energy formulation, and Lagrange’s equations while accounting for geometric nonlinearities, inertial coupling effects, and variations in the effective stiffness caused by structural deformation. First, the kinetic and strain energies of the mechanism components are derived, and the nonlinear equations of motion are obtained in terms of generalized coordinates. The equations are then nondimensionalized, and the system response is decomposed into a static equilibrium component and dynamic oscillations about the equilibrium position. Subsequently, the equations of motion are linearized around different equilibrium configurations to determine the natural frequencies and mode shapes of the mechanism, which are then compared with finite element results. The results demonstrate that the proposed analytical model accurately predicts the static response and modal characteristics of the system. To investigate nonlinear free vibrations, the method of multiple time scales is employed to derive the modulation equations and extract the backbone curves of the system. The results indicate that the natural frequency decreases as the vibration amplitude increases, revealing a softening-type nonlinear behavior induced by geometric nonlinearities, while the analytical predictions show good agreement with numerical solutions for small and moderate amplitudes.