Tensor-ring (TR) decomposition has recently attracted considerable attention in solving the low-rank tensor completion (LRTC) problem. However, due to an unbalanced unfolding scheme used during the update of core tensors, the conventional TR-based completion methods usually require a large TR rank to achieve the optimal performance, which leads to high computational cost in practical applications. To overcome this drawback, we propose a new method to exploit the low TR-rank structure in this article. Specifically, we first introduce a balanced unfolding operation called tensor circular unfolding, by which the relationship between TR rank and the ranks of tensor unfoldings is theoretically established. Using this new unfolding operation, we further propose an algorithm to exploit the low TR-rank structure by performing parallel low-rank matrix factorizations to all circularly unfolded matrices. To tackle the problem of nonuniform missing patterns, we apply a row weighting trick to each circularly unfolded matrix, which significantly improves the adaptive ability to various types of missing patterns. The extensive experiments have demonstrated that the proposed algorithm can achieve outstanding performance using a much smaller TR rank compared with the conventional TR-based completion algorithms; meanwhile, the computational cost is reduced substantially.