The metaphors of the Red Queen and the arms race have inspired a large amount of research aimed at understanding the process of antagonistic coevolution between hosts and parasites. One approach that has been heavily used is to estimate the strength of parasite local adaptation using a reciprocal cross infection or transplant study. These studies frequently conclude that the locally adapted species is ahead in the coevolutionary race. Here, I use mathematical models to decompose parasite infectivity into components attributable to local versus global adaptation and to develop a robust index of which species is ahead in the coevolutionary race, which I term coevolutionary advantage. Computer simulations of coevolving host-parasite interactions demonstrate that because the magnitudes of local and global adaptation are largely independent, the link between the sign of local adaptation and coevolutionary advantage is tenuous. A consequence of the weak coupling between local adaptation and coevolutionary advantage is that the bulk of existing empirical studies do not sample enough populations for any reliable conclusions to be drawn. Together, these results suggest that the long-standing conventional wisdom holding that locally adapted parasites are ahead in the coevolutionary race should be reconsidered.