Parsers that parametrize over wider scopes are generally more accurate than edge-factored models. For graph-based non-projective parsers, wider factorizations have so far implied large increases in the computational complexity of the parsing problem. This paper introduces a “crossing-sensitive” generalization of a third-order factorization that trades off complexity in the model structure (i.e., scoring with features over multiple edges) with complexity in the output structure (i.e., producing crossing edges). Under this model, the optimal 1-Endpoint-Crossing tree can be found in O(n^4) time, matching the asymptotic run-time of both the third-order projective parser and the edge-factored 1-Endpoint-Crossing parser. The crossing-sensitive third-order parser is significantly more accurate than the third-order projective parser under many experimental settings and significantly less accurate on none.