We introduce parallel symbolic algorithms for bisimulation minimisation, to combat the combinatorial state space explosion along three different paths. Bisimulation minimisation reduces a transition system to the smallest system with equivalent behaviour. We consider strong and branching bisimilarity for interactive Markov chains, which combine labelled transition systems and continuous-time Markov chains. Large state spaces can be represented concisely by symbolic techniques, based on binary decision diagrams. We present specialised BDD operations to compute the maximal bisimulation using signature-based partition refinement. We also study the symbolic representation of the quotient system and suggest an encoding based on representative states, rather than block numbers. Our implementation extends the parallel, shared memory, BDD library Sylvan, to obtain a significant speedup on multi-core machines. We propose the usage of partial signatures and of disjunctively partitioned transition relations, to increase the parallelisation opportunities. Also our new parallel data structure for block assignments increases scalability. We provide SigrefMC, a versatile tool that can be customised for bisimulation minimisation in various contexts. In particular, it supports models generated by the high-performance model checker LTSmin, providing access to specifications in multiple formalisms, including process algebra. The extensive experimental evaluation is based on various benchmarks from the literature. We demonstrate a speedup up to 95 for computing the maximal bisimulation on one processor. In addition, we find parallel speedups on a 48-core machine of another 17 for partition refinement and 24 for quotient computation. Our new encoding of the reduced state space leads to smaller BDD representations, with up to a 5162-fold reduction.