The interplay of magnetic and elastic properties due to geometrical frustration in antiferromagnetic Mn-aniperovskite nitrides manifests itself in a range of phenomena such as the barocaloric (BCE), piezomagnetic (PME), magnetovolume effect (MVE), and the related negative thermal expansion (NTE). This systematic computational study uses density functional theory across a wide range of cubic antiperovskites Mn$_3$AN (A = Rh, Pd, Ag, Co, Ni, Zn, Ga, In, Sn) in order to account for variations in the magnetic frustration based on features of the electronic structure. It focuses on PME - the linear dependence of magnetisation on applied biaxial strain. The PME in Mn$_3$SnN predicted here is an order of magnitude larger than PME modelled so far in Mn$_3$GaN,cite{lukashev2008theory} which opens the way to composite magnetoelectric effect in piezomagnetic/piezoelectric heterostructures. Moreover, the simulated PME as a zero temperature property is shown to be inversely proportional to the measured spontaneous volume expansion at a phase transition from paramagnetic (PM) to antiferromagnetic (AFM) state.cite{takenaka2014magnetovolume} On the fundamental level, such relation implies a significant suppression of spin fluctuations by the strong frustration in these systems. At the same time it can be used as a tool in search for materials with large negative thermal expansion and barocaloric effect.