Nestedness is a common pattern in interaction networks. However, its ecological and evolutionary meaning is under debate. Evidence shows that nestedness in mutualistic networks may be just a consequence of the species–abundance distribution. This has been questioned as abundance itself could be influenced by differences in generalism between species. Host-parasite networks in plant communities also show nested patterns, but their relationship with abundance has been seldom addressed. Importantly, an assessment of the potentially different effect of the number of interacting species (i.e. generalism levels) on the size of parasite and host populations can help understanding the role of abundance in determining both generalism and nestedness. Here we show that nestedness follows abundance expectations in an interaction network of climbing plants (i.e. structural parasites) and their tree and shrub hosts. Our results also point to a direct effect of abundance on both nestedness and generalism levels because species degree does not deviate from abundance expectations for both climbing plants and their hosts. Further, we found a similar level of discordance between generalization (a generalism measure independent of species abundance) and abundance for both parties. Our findings provide evidence that the factors underlying uneven abundance distributions can induce nestedness in interaction networks. We stress the importance of neutral processes related to species dominance as major determinants of nestedness in host-parasite networks.