Odd-parity magnetic multipole order, where time-reversal and space-inversion symmetries are spontaneously broken, may open a new paradigm in the fields of multipole physics, multiferroics, and spintronics. BaMnAs is a new candidate material. Here, the authors theoretically identify that the seemingly simple antiferromagnetic order in BaMnAs is actually magnetic hexadecapole order mixed with magnetic quadrupole order. Interestingly, doping hole carriers realize an itinerant odd-parity magnetic multipole order. Emergent electromagnetic responses, such as magnetoelectric effect, antiferromagnetic Edelstein effect, and electric current-induced nematicity, are clarified. In particular, the current-induced nematicity is a characteristic response of an itinerant magnetic hexadecapole state and indicates a novel phase, the “magnetopiezoelectric metal”.