Odd-parity magnetism represents an intriguing frontier in unconventional magnets; however, its realization has traditionally relied on noncollinear magnetic orders accompanied by broken spin conservation, which inevitably causes spin relaxation and dissipative transport of spin-encoded information. Here, we uncover a distinct route toward spin-conserving odd-parity magnon splitting enabled by antichiral Haldane-like flux (AHF) in collinear antiferromagnetic multiferroics. Symmetry analysis reveals that such AHF driven by intra-sublattice Dzyaloshinskii-Moriya interactions (DMI) gives rise to symmetry-dependent odd-parity magnon splittings, ranging from p-wave and f-wave to nodeless forms. Moreover, the coupling between ferroelectric modes and DMI provides an electric-field knob for manipulating chiral band splitting and magnon transport. Combining density functional theory (DFT) calculations, we identify a series of promising candidate materials in both 2D and bulk antiferromagnetic multiferroics. Our work provides new insights into the realization of odd-parity chiral magnons in collinear antiferromagnets and magnetoelectric coupling mechanisms in multiferroics.