We study an odd-parity magnetic multipole order in Ba1-xKxMn2As2 and related materials. Although BaMn2As2 is a seemingly conventional Mott insulator with G-type antiferromagnetic order, we identify the ground state as a magnetic hexadecapole ordered state accompanied by simultaneous time-reversal and space-inversion symmetry breaking. A symmetry argument and microscopic calculations reveal the ferroic ordering of leading magnetic hexadecapole moment and admixed magnetic quadrupole moment. Furthermore, we clarify electromagnetic responses characterizing the magnetic hexadecapole state of semiconducting BaMn2As2 and doped metallic systems. A magnetoelectric effect and antiferromagnetic Edelstein effect are shown. Interestingly, a counterintuitive current-induced nematic order occurs in the metallic state. The electric current along the z axis induces the xy-plane nematicity in sharp contrast to the spontaneous nematic order in superconducting Fe-based 122 compounds. Thus, the magnetic hexadecapole state of doped BaMn2As2 is regarded as a magnetopiezoelectric metal. Other candidate materials for magnetic hexadecapole order are proposed.