Unconventional magnets combining vanishing net magnetization with spin-split electronic structures and anomalous transport have attracted considerable attention. Using spin-group symmetry analysis, we derive general conditions for odd-parity spin splitting in two- and three-dimensional coplanar noncollinear antiferromagnets. We identify a key spin-group symmetry whose relativistic counterpart forbids time-reversal-odd transport and show that breaking it allows such transport in the presence of spin-orbit coupling. Tight-binding calculations validate these symmetry arguments and demonstrate an anomalous Hall effect in a fully compensated odd-parity magnetic state without an external field. By screening a three-dimensional magnetic materials database and constructing two-dimensional heterostructures, we further identify realistic platforms exhibiting pronounced spin-orientation-dependent anomalous Hall and magneto-optical effects. Our results establish a general symmetry framework for time-reversal-odd responses in odd-parity magnets and highlight fully compensated odd-parity magnets as a promising platform for controlling anomalous transport.