  The relationship between the eigenspectrum of Ising and XY quantum chains is
well known. Although the Ising model has a $Z(2)$ symmetry and the XY model a
$U(1)$ symmetry, both models are described in terms of free-fermionic
quasi-particles. The fermionic quasi-energies are obtained by means of a
Jordan-Wigner transformation. On the other hand, there exist in the literature
a huge family of $Z(N)$ quantum chains whose eigenspectra, for $N>2$, are given
in terms of free parafermions and they are not derived from the standard
Jordan-Wigner transformation. The first members of this family are the $Z(N)$
free-parafermionic Baxter quantum chains. In this paper we introduce a family
of XY models that beyond two-body also have $N$-multispin interactions.
Similarly to the standard XY model they have a $U(1)$ symmetry and are also
solved by the Jordan-Wigner transformation. We show that with appropriate
choices of the $N$-multispin couplings, the eigenspectra of these XY models are
given in terms of combinations of $Z(N)$ free-parafermionic quasi-energies. In
particular all the eigenenergies of the $Z(N)$ free-parafermionic models are
also present in the related free-fermionic XY models. The correspondence is
established via the identification of the characteristic polynomial which fixes
the eigenspectrum. In the $Z(N)$ free-parafermionic models the quasi-energies
obey an exclusion circle principle that is not present in the related
$N$-multispin XY models.
