We discover that a standard Fourier transformation intrinsically imparts a parity-dependent phase shift to optical vortices, yielding a phase difference of π between modes carrying odd and even topological charges. This phase shift, however, is not directly observable from the intensity profile of the transformed beam. To render it measurable, we develop a spin-selective Fourier transformer by co-designing propagation-induced dynamic phase and Pancharatnam-Berry geometric phase in a pair of hybrid-lenses. This device converts the parity-dependent phase shift into orthogonal polarization states via spin-orbit interaction, enabling non-destructive separation of both individual and superimposed orbital angular momentum (OAM) modes without the need for an interferometer. Our work establishes Fourier transformation as a new mechanism for parity-resolved vortex manipulation, offering an integrable platform for high-dimensional photonic applications.