We propose a parametric dictionary learning algorithm to design structured dictionaries that sparsely represent graph signals. We incorporate the graph structure by forcing the learned dictionaries to be concatenations of subdictionaries that are polynomials of the graph Laplacian matrix. The resulting atoms capture the main spatial and spectral components of the graph signals of interest, leading to adaptive representations with efficient implementations. Experimental results demonstrate the effectiveness of the proposed algorithm for the sparse approximation of graph signals.