Pathogens such as African swine fever virus (ASFV) are an increasing threat to global livestock production with implications for economic well-being and food security. Quantification of epidemiological parameters, such as transmission rates and latent and infectious periods, is critical to inform efficient disease control. Parameter estimation for livestock disease systems is often reliant upon transmission experiments, which provide valuable insights in the epidemiology of disease but which may also be unrepresentative of at-risk populations and incur economic and animal welfare costs. Routinely collected mortality data are a potential source of readily available and representative information regarding disease transmission early in outbreaks. We develop methodology to conduct exact Bayesian parameter inference from mortality data using reversible jump Markov chain Monte Carlo incorporating multiple routes of transmission (e.g. within-farm secondary and background transmission from external sources). We use this methodology to infer epidemiological parameters for ASFV using data from outbreaks on nine farms in the Russian Federation. This approach improves inference on transmission rates in comparison with previous methods based on approximate Bayesian computation, allows better estimation of time of introduction and could readily be applied to other outbreaks or pathogens.