  We discuss the quasi-normal oscillations of black holes which are sourced by
a nonlinear electrodynamic field. While previous studies have focused on the
computation of quasi-normal frequencies for the wave or higher spin equation on
a fixed background geometry described by such black holes, here we compute for
the first time the quasi-normal frequencies for the coupled
electromagnetic-gravitational linear perturbations.
  To this purpose, we consider a parametrized family of Lagrangians for the
electromagnetic field which contains the Maxwell Lagrangian as a special case.
In the Maxwell case, the unique spherically symmetric black hole solutions are
described by the Reissner-Nordstr\"om family and in this case it is well-known
that the quasi-normal spectra in the even- and odd-parity sectors are identical
to each other. However, when moving away from the Maxwell case, we obtain
deformed Reissner-Nordstr\"om black holes, and we show that in this case there
is a parity splitting in the quasi-normal mode spectra. A partial explanation
for this phenomena is provided by considering the eikonal (high-frequency)
limit.
