  Recent detections of gravitational waves have made black hole quasinormal
modes a powerful tool in testing predictions of general relativity.
Understanding the spectrum of these quasinormal modes in a broad class of
theories beyond general relativity and a variety of astrophysical environments
around black holes remains vital. In this work, we study the quasinormal mode
spectrum of parametrized deformations of a non-rotating black hole in the
vacuum. Following Vigeland and Hughes, we model these parametrized deformations
as axisymmetric multipole moments in the Weyl coordinates with amplitudes much
less than the amplitude of the Schwarzschild potential. These tiny bumps in the
black hole geometry satisfy the linearized vacuum Einstein equations and are
asymptotically flat. We use the recently developed modified Teukolsky formalism
to derive one decoupled differential equation for the radiative Weyl scalar
$\Psi_0$. We then use the eigenvalue perturbation method to compute the
quasinormal mode frequency shifts of both even- and odd-parity modes with
$\ell=2,3$ and up to the overtone number $n=2$ for the Weyl multipoles with
$\ell_W=2,3$. Our calculation provides an avenue to directly connect the
multipole moments of a modified black hole spacetime to the QNM frequency
shifts in a parametric way.
