Violation of parity symmetry gives rise to various physical phenomena such as nonlinear transport and cross-correlated responses. In particular, the nonlinear conductivity has been attracting a lot of attentions in spin-orbit coupled semiconductors, superconductors, topological materials, and so on. In this paper we present theoretical study of the nonlinear conductivity in odd-parity magnetic multipole ordered systems whose $mathcal{PT}$-symmetry is essentially distinct from the previously studied acentric systems. Combining microscopic formulation and symmetry analysis, we classify the nonlinear responses in the $mathcal{PT}$-symmetric systems as well as $mathcal{T}$-symmetric (non-magnetic) systems, and uncover nonlinear conductivity unique to the odd-parity magnetic multipole systems. A giant nonlinear Hall effect, nematicity-assisted dichroism and magnetically-induced Berry curvature dipole effect are proposed and demonstrated in a model for Mn-based magnets.
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