Various recent proposals increase the distinguishing power of Graph Neural Networks (GNNs) by propagating features between k-tuples of vertices. The distinguishing power of these “higher-order” GNNs is known to be bounded by the k-dimensional Weisfeiler-Leman (WL) test, yet their O(n^k) memory requirements limit their applicability. Other proposals infuse GNNs with local higher-order graph structural information from the start, hereby inheriting the desirable O(n) memory requirement from GNNs at the cost of a one-time, possibly non-linear, preprocessing step. We propose local graph parameter enabled GNNs as a framework for studying the latter kind of approaches and precisely characterize their distinguishing power, in terms of a variant of the WL test, and in terms of the graph structural properties that they can take into account. Local graph parameters can be added to any GNN architecture, and are cheap to compute. In terms of expressive power, our proposal lies in the middle of GNNs and their higher-order counterparts. Further, we propose several techniques to aide in choosing the right local graph parameters. Our results connect GNNs with deep results in finite model theory and finite variable logics. Our experimental evaluation shows that adding local graph parameters often has a positive effect for a variety of GNNs, datasets and graph learning tasks.