  A set of tidal Love numbers quantifies tidal deformation of compact objects
and is a detectable imprint in gravitational waves from inspiralling binary
systems. The measurement of black hole Love numbers allows to test strong-field
gravity. In this paper, we present a parametrized formalism to compute the Love
numbers of static and spherically symmetric black hole backgrounds, connecting
the underlying equations of a given theory with detectable quantities in
gravitational-wave observations in a theory-agnostic way. With this formalism,
we compute the Love numbers in several systems. We further classify black hole
Love numbers according to whether they vanish, are nonzero, or are ``running''
(scale-dependent), in theories or backgrounds that deviate perturbatively from
the GR values. The construction relies on static linear perturbations and
scattering theory. Our analytic and numerical results are in excellent
agreement. As a side result, we show how to use Chandrasekhar's relations to
relate basis of even parity to odd parity.
