Existing theory of host-parasite interactions has identified the genetic specificity of interaction as a key variable affecting the outcome of coevolution. The Matching Alleles (MA) and Gene For Gene (GFG) models have been extensively studied as the canonical examples of specific and non-specific interaction. The generality of these models has recently been challenged by uncovering real-world host-parasite systems exhibiting specificity patterns that fit neither MA nor GFG, and by the discovery of symbiotic bacteria protecting insect hosts against parasites. In the present paper we address both challenges, simulating a large number of non-canonical models of host-parasite interactions that explicitly incorporate symbiont-based host resistance. To assess the genetic specialisation in these hybrid models, we develop a quantitative index of specificity applicable to any coevolutionary model based on a fitness matrix. We find qualitative and quantitative effects of host-parasite and symbiont-parasite specificities on genotype frequency dynamics, allele survival, and mean host and parasite fitnesses.