In the present paper, we propose a new definition of discrete parabolas, the so-called arithmetic discrete parabolas. We base our approach on a non-constant thickness function and characterized the 0-connected and 1-connected parabolas in terms of thickness function. This results extend the well-known characterization of the κ-connectedness of arithmetic discrete lines, depending on the norm || ·||∞ and || ·||1 of their normal vector.