  In this paper we study slant null curves with respect to the original
parameter on 3-dimensional normal almost contact B-metric manifolds with
parallel Reeb vector field. We prove that for non-geodesic such curves there
exists a unique Frenet frame for which the original parameter is distinguished.
Moreover, we obtain a necessary condition this Frenet frame to be a Cartan
Frenet frame with respect to the original parameter. Examples of the considered
curves are constructed.
