We describe a modular apparatus for use in parity-violation measurements in epithermal neutron-nucleus resonances with high instantaneous neutron fluxes at the Manuel Lujan Jr. Neutron Scattering Center at Los Alamos National Laboratory. This apparatus is designed to conduct high-precision measurements of the parity-odd transmission asymmetry of longitudinally polarized neutrons through targets containing nuclei with p-wave neutronnucleus resonances in the 0.1-10 eV energy regime and to accommodate a future search for time reversal violation in polarized neutron transmission through polarized nuclear targets. The apparatus consists of an adjustable neutron and gamma collimation system, a 3 He-4 He ion chamber neutron flux monitor, two identical cryostats for target cooling, an adiabatic eV-neutron spin flipper, a near-unit efficiency 6 Li-7 Li scintillation detector operated in current mode, a flexible CAEN data acquisition system, and a neutron spin filter based on spin-exchange optical pumping of 3 He gas. We describe the features of the apparatus design devoted to the suppression of systematic errors in parity-odd asymmetry measurements. We describe the configuration of the apparatus used to conduct a precision measurement of parity violation at the 0.7 eV p-wave resonance in 139 La which employs two identical 139 La targets, one to polarize the beam on the p-wave resonance using the weak interaction and one to analyze the polarization.