  An optical ring cavity is used to place the first laboratory constraints on
parity-odd nonrenormalizable Lorentz violation. Variations in resonant
frequencies are limited to parts in $10^{15}$. Absolute sensitivity to
Lorentz-violating operators of mass dimension 6 is improved by a factor of a
million over existing parity-even microwave-cavity bounds. Sensitivity to
dimension-8 violations is improved by fourteen orders of magnitude.
