NiPS₃ is a layered van der Waals magnet that provides a unique platform for exploring the interplay between excitons and magnetic order. In its antiferromagnetic phase, strong many-body interactions give rise to a many-body exciton, which manifests itself as an exceptionally sharp resonance near 1.47 eV in optical absorption spectra and photoluminescence. Previous theoretical studies have assigned the local ground state and the many-body exciton to spin-triplet and spin-singlet states, respectively, both with even parity. This picture, however, cannot account for the observed optical visibility of the many-body exciton because one-photon transitions between these states are forbidden by spin and parity selection rules. Here, using group-theoretical analysis and exact diagonalization of single- and two-cluster models, we show that zigzag antiferromagnetic order breaks the relevant symmetries and activates the otherwise forbidden transition. We further find that trigonal distortion of the local ligand environment and the inter-cluster exchange interaction relax additional spatial and spin constraints on the transition, producing an exciton signal in the optical conductivity that is distinguishable from the spectral background. These results provide a microscopic explanation for the optical visibility of the many-body exciton and its connection to antiferromagnetic order in NiPS₃.