A (quasirelativistic) two-component density functional theory (DFT) approach to the computation of parity-violating energy differences between enantiomers is presented which is based on the zeroth-order regular approximation (ZORA). This approach is employed herein to compute parity-violating energy differences between several P and M conformations of dihydrogen dichalcogenides (H2X2 with X=O, S, Se, Te, Po), of which some compounds have recently been suggested as potential molecular candidates for the first experimental measurement of parity-violating effects in chiral molecules. The DFT ZORA results obtained in this work with "pure" density functionals are anticipated to deviate by well less than 1% from data that would be computed within related (relativistic) four-component Dirac-Kohn-Sham-Coulomb schemes. In our implementation of the ZORA slightly larger relative deviations are expected for hybrid functionals, depending on the amount of "exact" exchange. For B3LYP (20% exact exchange) differences are estimated to amount to at most 3% in hydrogen peroxide, 2% in disulfane, and 1% or less for the heavier homologs. Thus, the present two-component approach is expected to perform excellently when compared to four-component density functional schemes while being at the same time computationally more efficient. The ZORA approach will therefore be of particular interest for the prediction of parity-violating vibrational frequency shifts, for instance, in isotopomers of H(2)Se(2) and H(2)Te(2).