This letter investigates the three-dimensional (3D) downlink sparse channel estimation for tethered aerial platform-enabled multi-user communication systems operating in a frequency division duplexing mode with large-scale antenna arrays. To this end, we design a non-identical Bernoulli-Gaussian distribution-based channel model that reflects the potential common sparsity caused by distant scatterers in a low-altitude environment. A low-complexity message passing-based channel estimation algorithm without the need for prior knowledge of channel sparsity is proposed. Specifically, this algorithm leverages a sparsity rate-aided adaptive majority voting scheme to automatically identify diverse kinds of channel supports. It also utilizes an improved sparse channel estimator to recover sparse channels accurately with the achieved channel supports. Compared with its existing counterparts, the proposed algorithm achieves better estimation performance under various parameter settings and approaches the performance limit with perfect knowledge of the channel common supports.