  We study an odd-parity magnetic multipole order in
Ba$_{1-x}$K$_x$Mn$_2$As$_2$ and related materials. Although BaMn$_2$As$_2$ 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 BaMn$_2$As$_2$ and doped metallic systems.
A magnetoelectric effect and antiferromagnetic Edelstein effect are shown.
Interestingly, a counter-intuitive currentinduced nematic order occurs in the
metallic state. The electric current along the \textit{z}-axis induces the
\textit{xy}-plane nematicity in sharp contrast to the spontaneous nematic order
in superconducting Febased 122-compounds. Thus, the magnetic hexadecapole state
of doped BaMn$_2$As$_2$ is regarded as a magnetopiezoelectric metal. The
anomalous responses stem from the peculiar symmetry of the parity-violating
magnetic order. Other candidate materials for magnetic hexadecapole order are
proposed.
