This paper addresses arguments due to Dyła (1984) that well-formed across-the-board dependencies must respect a parallelism requirement in both abstract Case and morphological case. New data from Polish and Russian are presented to suggest that the abstract Case requirement is more appropriately replaced by one formulated in terms of thematic prominence. Next, it is shown that both the morphological compatibility and thematic prominence constraints also apply to parasitic gap constructions. It is argued that this result follows from the idea that across-the-board and parasitic gap extraction both involve null operators. Finally, further similarities and differences between the two construction types are discussed.