In this study, we make the first contribution to investigating conditions under which normal almost paracontact metric manifold of dimension 3 has cyclic parallel Ricci tensor, η-parallel Ricci tensor, Ricci-semisymmetry and locally φ-symmetry. In the end an example of a 3-dimensional normal almost paracontact metric manifold which is locally φ-symmetric and has cyclic parallel Ricci tensor is presented.