Local dynamics of para-substituted poly(1-chloro-2-phenylacetylene) (PCPA) derivatives was studied using quasielastic neutron scattering. Broadening of the dynamic scattering law (S(Q,omega)) was clearly observed for para-substituted PCPA derivatives even in the glassy state, and the observed S(Q,omega) was well fitted to a sum of narrow and broad Lorentzians, which represent the slow and fast motions, respectively. The half-width at half-maximum (Gamma) of the narrow Lorentzian exhibited a positive correlation with the CO2 and CH4 permeability coefficients (P), while the Gamma from the broad Lorentzian hardly changed despite of the variation of permeability coefficient, indicating that only the narrow component (i.e., the slow motion) contributes to the gas permeability. The Gamma from the narrow component was approximately proportional to the diffusion coefficient (D) of the CO2 and CH4 gases, whereas it did not correlate with solubility coefficient (S). This implies that the local mobility affects the gas permeability through the diffusion term. The slow motion with a time scale of several tens of picoseconds seems to play an essential role for the gas permeability in glassy polymers.