To achieve collaborative optimization across different types of virtual power plants (VPPs), this paper proposes a distributed optimal operation strategy for multi-type VPP cooperative games based on the minimum-maximum regret (MMR) theory. First, a cooperative game model for hybrid VPPs, load-type VPPs and generation-type VPPs is established using Nash bargaining theory, which is further decomposed into two computationally tractable subproblems to solve for optimal payoffs and implement profit allocation. Next, a communication-fault-tolerant parallel alternating direction method of multipliers is adopted to solve the two subproblems, which not only boosts computational efficiency but also strengthens algorithmic stability in the presence of low-to-medium communication failure probabilities. Then, by evaluating the regret values of different scenarios under the MMR framework, an optimal scheduling strategy that balances robustness and economic performance is identified. Finally, the effectiveness of the proposed model and method is validated via simulation case studies.