Bayesian networks and their variants are widely used for modelling gene regulatory and protein signalling networks. In many settings, it is the underlying network structure itself that is the object of inference. Within a Bayesian framework inferences regarding network structure are made via a posterior probability distribution over graphs. However, in practical problems, the space of graphs is usually too large to permit exact inference, motivating the use of approximate approaches. An MCMC-based algorithm known as MC3 is widely used for network inference in this setting. We argue that recent trends towards larger sample size datasets, while otherwise advantageous, can, for reasons related to concentration of posterior mass, render inference by MC3 harder. We therefore exploit an approach known as parallel tempering to put forward an algorithm for network inference which we call MC4. We show empirical results on both synthetic and proteomic data which highlight the ability of MC4 to converge faster and thereby yield demonstrably accurate results, even in challenging settings where MC3 fails.