The support structure of the moving mirror constitutes the most crucial component of the high-precision Michelson interferometer, exerting a decisive impact on the quality of the interference signal. Conventional moving mirror support is accomplished through mechanical guide rails or magnetic levitation in combination with a drive motor, which is expensive, difficult to maintain, has a low life span, and has low guiding accuracy under the load of large mass and large stroke angle mirrors. To address these issues, a flexible support structure for the moving mirror of the Michelson interferometer is designed, featuring low cost, high precision, large load capacity, and large travel range. In this paper, the working principle of the Michelson interferometer is presented, and the impacts of the travel of the moving mirror, guiding accuracy, and velocity uniformity on the accuracy and stability of the interferometer are analyzed. The influence of the parasitic displacement perpendicular to the direction of motion on the parasitic path difference is quantified when the angle mirror is employed as the moving mirror. With the parallelogram guiding structure serving as the basic prototype, four parallelogram structures are nested both internally and externally to form the fundamental framework of the support structure, effectively amplifying the motion stroke of the entire structure. The flexible support structure is arranged symmetrically to counteract the parasitic displacement in the horizontal direction. Taking the load capacity, travel, and guiding accuracy of the flexible support structure as the optimization targets, the structural stiffness model is proposed. Based on the Awtar beam constraint model, the force-displacement relationship of the flexible reed with unilateral constraint is derived from the deformation mechanism of the cantilever beam, and subsequently, the stiffness formula of the flexible reed with unilateral constraint is deduced. The stiffness model of two unilateral constrained flexible reeds is obtained by means of Hooke's law. On this basis, the stiffness model of the entire flexible structure is derived in accordance with the series-parallel relationship of the flexible reeds in the flexible support structure. The finite element method combined with the stiffness model of the flexible structure is utilized to optimize the size parameters of the structure, and the transition fillet is set at the connection of the flexible reed and the rigid part to mitigate the stress concentration phenomenon. By comparing the ratios of yield strength to elastic modulus of different materials, 7075 aluminum alloy is determined as the optimal material for the structure, and the three-dimensional model design and simulation of the flexible structure are conducted. The finite element analysis results indicate that the maximum tensile stress of the flexible structure amounts to 169 MPa, the structural safety margin is 1.98, and the parasitic displacement perpendicular to the movement direction is less than 4.1 μm when the travel attains 4.5 mm under the load of 1.5 kg (angle mirror). A special test platform was established by means of a spiral micrometer, a high-precision grating scale meter, a digital dynamometer, and weights. The force-displacement relationship and parasitic displacement of the test pieces were examined, and the errors between the simulation results and the experimental results were analyzed. The results demonstrate that the parasitic displacement perpendicular to the motion direction is less than 3.2 μm and 4.7 μm, and the root-mean-square error of straightness is 0.96 μm and 1.5 μm respectively when the flexible support structure is subjected to 0.5 kg and 1.5 kg load (angle mirror) within the range of 4.5 mm travel. The test results of this structure are consistent with the design results, and can meet the support requirements of the satellite-borne Michelson interferometer for high-precision linear moving mirrors. It can also be used in other motion systems that require large stroke, large load, long life, and high guiding accuracy.