A search for Lorentz violation in electrodynamics was performed by measuring the resonant frequency difference between two counterpropagating directions of an optical ring cavity. Our cavity contains a dielectric element, which makes our cavity sensitive to the violation. The laser frequency is stabilized to the counterclockwise resonance of the cavity, and the transmitted light is reflected back into the cavity for resonant frequency comparison with the clockwise resonance. This double-pass configuration enables a null experiment and gives high common mode rejection of environmental disturbances. We found no evidence for odd-parity anisotropy at the level of δc/c ≲ 10(-14). Within the framework of the standard model extension, our result put more than 5 times better limits on three odd-parity parameters κ(o+)(JK) and a 12 times better limit on the scalar parameter κ(tr) compared with the previous best limits.