We study a mechanism of generating the trispectrum (4-point correlation) of curvature perturbation through the dynamics of a spectator axion field and U(1) gauge field during inflation. Owing to the Chern-Simons coupling, only one helicity mode of the gauge field experiences a tachyonic instability and sources scalar perturbations. Sourced curvature perturbation exhibits parity-violating nature which can be tested through its trispectrum. We numerically compute the parity-even and parity-odd components of the sourced trispectrum. It is found that the ratio of parity-odd to parity-even mode can reach O(10%) in an exact equilateral momentum configuration. We also investigate a quasi-equilateral shape where only one of the momenta is slightly longer than the other three, and find that the parity-odd mode can reach, and more interestingly, surpass the parity-even one. This may help us to interpret a large parity-odd trispectrum signal extracted from BOSS galaxy-clustering data.