Water molecules confined between biological membranes exhibit a distinctive non-Gaussian displacement distribution, far different from that of bulk water. Here, we introduce a new transport equation for water molecules in the intermembrane space, quantitatively explaining molecular dynamics simulation results. We find that the unique transport dynamics of water molecules stems from the lateral diffusion coefficient fluctuation caused by their longitudinal motion in the direction perpendicular to the membranes. We also identify an interfacial region where water possesses distinct physical properties, which is unaffected by changes in the intermembrane separation.