  Black-hole spectroscopy is arguably the most promising tool to test gravity
in extreme regimes and to probe the ultimate nature of black holes with
unparalleled precision. These tests are currently limited by the lack of a
ringdown parametrization that is both robust and accurate. We develop an
observable-based parametrization of the ringdown of spinning black holes beyond
general relativity, which we dub ParSpec (Parametrized Ringdown Spin Expansion
Coefficients). This approach is perturbative in the spin, but it can be made
arbitrarily precise (at least in principle) through a high-order expansion. It
requires O(10) ringdown detections, which should be routinely available with
the planned space mission LISA and with third-generation ground-based
detectors. We provide a preliminary analysis of the projected bounds on
parametrized ringdown parameters with LISA and with the Einstein Telescope, and
discuss extensions of our model that can be straightforwardly included in the
future.
