Measurements of nuclear spin-dependent parity-violating (NSD-PV) effects provide an excellent opportunity to test nuclear models and to search for physics beyond the Standard Model. Molecules possess closely spaced states with opposite parity which may be easily tuned to degeneracy to greatly enhance the observed parity-violating effects. A high-sensitivity measurement of NSD-PV effects using light triatomic molecules is in preparation [E. B. Norrgard et al., Common. Phys. 2, 77 (2019)]. Importantly, by comparing these measurements in light nuclei with prior and ongoing measurements in heavier systems, the contribution to NSD-PV from Z(0)-boson exchange between the electrons and the nuclei may be separated from the contribution of the nuclear anapole moment. Furthermore, light triatomic molecules offer the possibility to search for new particles, such as the postulated Z boson. In this work, we detail a sensitive measurement scheme and present high-accuracy molecular and nuclear calculations needed for interpretation of NSD-PV experiments on triatomic molecules composed of light elements, Be, Mg, N, and C. The ab initio nuclear structure calculations, performed within the no-core shell model provide a reliable prediction of the magnitude of different contributions to the NSD-PV effects in the four nuclei. These results differ significantly from the predictions of the standard single-particle model and highlight the importance of including many-body effects in such calculations. In order to extract the NSD-PV contributions from measurements, a parity-violating interaction parameter W-PV, which depends on the molecular structure, needs to be known with a high accuracy. We have calculated these parameters for the triatomic molecules of interest using the relativistic coupled-cluster approach. In order to facilitate the interpretation of future experiments we provide uncertainties on the calculated parameters. A scheme for measurement using laser-cooled polyatomic molecules in a molecular fountain is presented, along with an estimate of the expected sensitivity of such an experiment. This experimental scheme, combined with the presented state-of-the-art calculations, opens exciting prospects for a measurement of the anapole moment and the PV effects due to the electron-nucleon interactions with unprecedented accuracy and for a new path towards detection of signatures of physics beyond the Standard Model.