  The entanglement asymmetry is a novel quantity that, using entanglement
methods, measures how much a symmetry is broken in a part of an extended
quantum system. So far it has only been used to characterise the breaking of
continuous Abelian symmetries. In this paper, we extend the concept to cyclic
$\mathbb{Z}_N$ groups. As an application, we consider the XY spin chain, in
which the ground state spontaneously breaks the $\mathbb{Z}_2$ spin parity
symmetry in the ferromagnetic phase. We thoroughly investigate the
non-equilibrium dynamics of this symmetry after a global quantum quench,
generalising known results for the standard order parameter.
