High-precision displacement platforms are of great significance in micro/nanoscale manipulation, ultraprecision manufacturing, biological manipulation, and scanning probe microscopes. This paper presents a novel compliant planar parallelogram mechanism (CPPM) for high-precision translational motion utilizing 8 flexible initially curved beams (ICBs). Compared with the existing high-precision displacement platforms for traditional motion, CPPM combines the characteristics of large motion ranges, bidirectional anti-buckling and high load-bearing. Moreover, this new mechanism introduces more design parameters compared to traditional parallelogram mechanisms, thus generating other desired characteristics by optimizing these parameters. In this paper, a general mathematical model is first derived for quick design and model-based optimization, followed by nonlinear finite element method (FEM) validation. Three desired performance characteristics in compliant translational mechanisms are proposed thereafter to obtain three optimized designs through global optimization of the preset geometric parameters. Experimental results verify the accuracy of the mathematical model and demonstrate centimeter-level high-precision positioning within a large motion range. Finally, a comparative discussion of the anti-buckling mechanisms for translation is presented.