Parallel treebanks provide a systematic way of expressing the structural relationships between source and target texts. In this paper, we present the general design principles behind the Copenhagen Dependency Treebanks, a set of parallel treebanks for Danish, English, German, Italian and Spanish with a unified annotation of morphology, syntax, discourse, and tranlational equivalence. Finally, we suggest some hypotheses about morphology and discourse, and describe how we plan to explore them empirically on the basis of the treebanks.