The Perdew-Zunger self-interaction correction (SIC) was implemented self-consistently within a molecular density functional theory (DFT) program, using the Krieger-Li-Iafrate approximation to the optimized effective potential, and the Vosko-Wilk-Nusair (VWN) functional. The computationally efficient implementation relies on the fitting of orbital densities for the evaluation of orbital Coulomb potentials, and allows for routine applications to large molecules. Due to the use of the effective potential approach, the evaluation of the energy derivatives can be handled by standard Kohn-Sham DFT techniques in a straightforward way. The SIC-VWN technique is applied to the calculation of nuclear magnetic resonance (NMR) parameters in representative small molecules, containing C, H, N, O, and F. Removal of self-interaction leads to a substantial improvement in the calculated isotropic chemical shifts for N, O, and F, where SIC-VWN holds an advantage over both local (VWN), and gradient-corrected functionals (Becke 88-Perdew 86, BP86). For C and H isotropic chemical shifts, which are well described by the gradient-corrected functionals, SIC-VWN performs as well as BP86. SIC-VWN also improves the description of the absolute chemical shielding, and of the principal components of the NMR shielding tensors. The changes arise mainly from adjustments in the Kohn-Sham orbital energies, leading to a better description of the paramagnetic contribution to the shielding tensor. For spin-spin coupling constants, SIC-VWN improves the description of the paramagnetic contribution. At the same time, the magnitude of the Fermi contact term is underestimated, yielding mixed overall results. Slow convergence of the spin-spin coupling results with the basis set size prevents a conclusive statistical evaluation for this property. The clear physical origin of the SIC-VWN effect in the prediction of magnetic properties opens the tantalizing possibility that this technique may be effective in solving problems often encountered in the calculations of NMR parameters of heavier nuclei.