  Each quiver appearing in a seed of a cluster algebra determines a
corresponding group, which we call a cluster group, which is defined via a
presentation. Grant and Marsh showed that, for quivers appearing in seeds of
cluster algebras of finite type, the associated cluster groups are isomorphic
to finite reflection groups. As for finite Coxeter groups, we can consider
parabolic subgroups of cluster groups. We prove that, in the type $A_{n}$ case,
there exists an isomorphism between the lattice of subsets of the defining
generators of the cluster group and the lattice of its parabolic subgroups.
Moreover, each parabolic subgroup has a presentation given by restricting the
presentation of the whole group.
