We explore the contribution of lexical and inflectional morphology features to dependency parsing of Arabic, a morphologically rich language with complex agreement patterns. Using controlled experiments, we contrast the contribution of different part-of-speech (POS) tag sets and morphological features in two input conditions: machine-predicted condition (in which POS tags and morphological feature values are automatically assigned), and gold condition (in which their true values are known). We find that more informative (fine-grained) tag sets are useful in the gold condition, but may be detrimental in the predicted condition, where they are outperformed by simpler but more accurately predicted tag sets. We identify a set of features (definiteness, person, number, gender, and undiacritized lemma) that improve parsing quality in the predicted condition, whereas other features are more useful in gold. We are the first to show that functional features for gender and number (e.g., “broken plurals”), and optionally the related rationality (“humanness”) feature, are more helpful for parsing than form-based gender and number. We finally show that parsing quality in the predicted condition can dramatically improve by training in a combined gold+predicted condition. We experimented with two transition-based parsers, MaltParser and Easy-First Parser. Our findings are robust across parsers, models, and input conditions. This suggests that the contribution of the linguistic knowledge in the tag sets and features we identified goes beyond particular experimental settings, and may be informative for other parsers and morphologically rich languages.