Continuous Distributed Constraint Optimization Problem (C-DCOPs) are a powerful framework to model problems with continuous variables in multi-agent systems. Previous works about C-DCOP algorithms mainly use pseudo-trees to guarantee the anytime property or are without anytime property guarantees. However, there is a risk of privacy leakage in the pseudo-tree due to utility passing between agents. Therefore, based on the basic constraint graph instead of the pseudo-tree, we (i) extend the Maximum Gain Message (MGM) algorithm by combining the local search strategy to solve C-DCOPs, named Continuous MGM (C-MGM), and it’s able to guarantee the monotonicity of the solution quality; (ii) propose a Parallel C-MGM (C-PMGM) algorithm to improve the solution quality through parallel random search; and (iii) introduce the differential search into C-PMGM to design a Parallel Differential Search C-MGM (C-PDSM) algorithm, which constructs a heuristic method to speed up convergence and improve solution quality. Compared to other anytime C-DCOP algorithms using pseudo-trees, the proposed three algorithms can exhibit better performance in avoiding privacy violations. We theoretically prove that the proposed algorithms are the anytime algorithms and empirically demonstrate that our algorithms outperform the state-of-the-art C-DCOP algorithms.