Most morphologically rich languages with free word order use case systems to mark the grammatical function of nominal elements, especially for the core argument functions of a verb. The standard pipeline approach in syntactic dependency parsing assumes a complete disambiguation of morphological (case) information prior to automatic syntactic analysis. Parsing experiments on Czech, German, and Hungarian show that this approach is susceptible to propagating morphological annotation errors when parsing languages displaying syncretism in their morphological case paradigms. We develop a different architecture where we use case as a possibly underspecified filtering device restricting the options for syntactic analysis. Carefully designed morpho-syntactic constraints can delimit the search space of a statistical dependency parser and exclude solutions that would violate the restrictions overtly marked in the morphology of the words in a given sentence. The constrained system outperforms a state-of-the-art data-driven pipeline architecture, as we show experimentally, and, in addition, the parser output comes with guarantees about local and global morpho-syntactic wellformedness, which can be useful for downstream applications.