The determination of nuclear symmetry energy, and in particular, its density dependence, is a long-standing problem for nuclear physics community. Previous studies have found that the product of electric dipole polarizability $alpha_D$ and symmetry energy at saturation density $J$ has a strong linear correlation with $L$, the slope parameter of symmetry energy. However, current uncertainty of $J$ hinders the precise constraint on $L$. We investigate the correlations between electric dipole polarizability $alpha_D$ (or times symmetry energy at saturation density $J$) in Sn isotopes and the slope parameter of symmetry energy $L$ using the quasiparticle random-phase approximation based on Skyrme Hartree-Fock-Bogoliubov. A strong and model-independent linear correlation between $alpha_D$ and $L$ is found in neutron-rich Sn isotopes where pygmy dipole resonance (PDR) gives a considerable contribution to $alpha_D$, attributed to the pairing correlations playing important roles through PDR. This newly discovered linear correlation would help one to constrain $L$ and neutron-skin thickness $Delta R_textnormal{np}$ stiffly if $alpha_D$ is measured with high resolution in neutron-rich nuclei. Besides, a linear correlation between $alpha_D J$ in a nucleus around $beta$-stability line and $alpha_D$ in a neutron-rich nucleus can be used to assess $alpha_D$ in neutron-rich nuclei.
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