Organic materials are promising for applications in spintronics due to their long spin-relaxation times in addition to their chemical flexibility and relatively low production costs. Most studies of organic materials for spintronics focus on nonpolar dielectrics or semiconductors, serving as passive elements in spin transport devices. Here, we demonstrate that employing organic ferroelectrics, such as poly(vinylidene fluoride) (PVDF), as barriers in magnetic tunnel junctions (MTJs) allows new functionality in controlling the tunneling spin polarization via the ferroelectric polarization of the barrier. Using first-principles methods based on density functional theory we investigate the spin-resolved conductance of Co/PVDF/Co and Co/PVDF/Fe/Co MTJs as model systems. We show that these tunnel junctions exhibit multiple resistance states associated with different magnetization configurations of the electrodes and ferroelectric polarization orientations of the barrier. Our results indicate that organic ferroelectrics may open a new and promising route in organic spintronics with implications for low-power electronics and nonvolatile data storage.