We use inelastic neutron scattering to show that superconductivity in electron-underdoped NaFe_{0.985}Co_{0.015}As induces a dispersive sharp resonance near E_{r1}=3.25 meV and a broad dispersionless mode at E_{r2}=6 meV. However, similar measurements on overdoped superconducting NaFe_{0.935}Co_{0.045}As find only a single sharp resonance at E_{r}=7 meV. We connect these results with the observations of angle-resolved photoemission spectroscopy that the superconducting gaps in the electron Fermi pockets are anisotropic in the underdoped material but become isotropic in the overdoped case. Our analysis indicates that both the double neutron spin resonances and gap anisotropy originate from the orbital dependence of the superconducting pairing in the iron pnictides. Our discovery also shows the importance of the inelastic neutron scattering in detecting the multiorbital superconducting gap structures of iron pnictides.