  Modifications to general relativity lead to effects in the spectrum of
quasi-normal modes of black holes. In this paper, we develop a parametrized
formalism to describe deviations from general relativity in the Teukolsky
equation, which governs linear perturbations of spinning black holes. We do
this by introducing a correction to the effective potential of the Teukolsky
equation in the form of a $1/r$ expansion controlled by free parameters. The
method assumes that a small deviation in the effective potential induces a
small modification in the spectrum of modes and in the angular separation
constants. We isolate and compute the universal linear contribution to the
quasi-normal mode frequencies and separation constants in a set of
coefficients, and test them against known examples in the literature (massive
scalar field, Dudley-Finley equation and higher-derivative gravity). We make
the coefficients publicly available for relevant overtone, angular momentum and
azimuthal numbers of modes and different values of the black hole spin.
