This paper considers optically inactive molecules possessing a symmetry plane. Degenerated excited electronic states in such molecules may, in principle, differ in symmetry with respect to mapping onto the symmetry plane. Should this prove the case, the parity-nonconserving electron-nuclear interaction (PNI) causes the degenerated electronic level to suffer a splitting linear in the Weinberg constant. The paper analyzes from this standpoint the lowest-lying excited states in the ten-electron HF, NH3, B-2, and H2O molecules. Two of them, namely HF and NH3, possess the necessary and sufficient symmetry properties for such a linear splitting to occur. Factors are discussed that augment the PNI-induced splitting of the excited states under consideration in comparison with the splitting of the ground state in left- and right-handed modifications of optically active molecules. Computations confirm the occurrence of a great (approximate to 10(-13) eV) splitting of the levels being considered due to the PNI effect. A similar effect can also occur in the electronic ground state of paramagnetic molecules, such as NO. The computation uses the consistent multiple-electron perturbation theory with a model single-electron central field bare potential. The computer code used is a modification of the original code developed for precision atomic calculations. All the computations boil down to the solution of a single set of ordinary differential equations, i.e., a unidimensional procedure.