We study affine immersions, as introduced by Nomizu and Pinkall, of $M^n$ into $\R^{n+p}$. We call $M^n$ linearly full if the image of $M$ is not contained in a lower dimensional affine space. Typical examples of affine immersions are the Euclidean and semi-Riemannian immersions. A classification, under an additional assumption that the rank of the second fundamental form is at least two, of the hypersurfaces with parallel second fundamental form was obtained by Nomizu and Pinkall. If we assume that the second fundamental form is parallel and $M$ is linearly full, then $p \le n(n+1)/2$. In this paper we completely classify the affine immersions with parallel second fundamental form in $\R^{n+n(n+1)/2}$, obtaining amongst others the generalized Veronese immersions.