Robustness, defined as tolerance to perturbations such as mutations and environmental fluctuations, is pervasive in biological systems. However, robustness often coexists with its counterpart, evolvability--the ability of perturbations to generate new phenotypes. Previous models of gene regulatory network evolution have shown that robustness evolves under stabilizing selection, but it is unclear how robustness and evolvability will emerge in common coevolutionary scenarios. We consider a two-species model of coevolution involving one host and one parasite population. By using two interacting species, key model parameters that determine the fitness landscapes become emergent properties of the model, avoiding the need to impose these parameters externally. In our study, parasites are modeled on species such as cuckoos where mimicry of the host phenotype confers high fitness to the parasite but lower fitness to the host. Here, frequent phenotype changes are favored as each population continually adapts to the other population. Sensitivity evolves at the network level such that point mutations can induce large phenotype changes. Crucially, the sensitive points of the network are broadly distributed throughout the network and continually relocate. Each time sensitive points in the network are mutated, new ones appear to take their place. We have therefore named this phenomenon whack-a-mole sensitivity, after a popular fun park game. We predict that this type of sensitivity will evolve under conditions of strong directional selection, an observation that helps interpret existing experimental evidence, for example, during the emergence of bacterial antibiotic resistance.