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light of the recently discovered connection between grey-body factors and quasi normal ringing, we derive analytic expressions for the grey-body factors of generic parametrized spherically symmetric and asymptotically flat black holes. These expressions are presented as expansions in terms of the inverse multipole number and the coefficients of the parametrization. The obtained analytic formulas serve as good approximations whenever the deviation from the Schwarzschild geometry is not very large. We demonstrate that the primary parameter determining the grey-body factors is the deviation of the event horizon radius from its Schwarzschild value, while the higher-order coefficients of the parametrization, which govern the near-horizon geometry, are much less significant. This finding is consistent with recent observations that grey-body factors are considerably more stable against small deformations of the near-horizon geometry than quasinormal modes.