The simple incidence structure $𝒟(𝒜,2)$ formed by points and unordered pairs of distinct parallel lines of a finite affine plane $𝒜=(𝒫,ℒ)$ of order $n>2$ is a $2-(n^2,2n,2n-1)$ design. If $n=3$, $𝒟(𝒜,2)$ is the complementary design of $𝒜$. If $n=4$, $𝒟(𝒜,2)$ is isomorphic to the geometric design $A{G}_3(4,2)$ (see [2; Theorem 1.2]). In this paper we give necessary and sufficient conditions for a $2-(n^2,2n,2n-1)$ design to be of the form $𝒟(𝒜,2)$ for some finite affine plane $𝒜$ of order $n>4$. As a consequence we obtain a characterization of small designs $𝒟(𝒜,2)$.