This paper proposes a novel tensor-train deep neural network (TT-DNN) based channel estimator to tackle challenges of time-varying channel estimation in multiple-input multiple-output (MIMO) systems. A centralized DNN channel estimator can be realized by a distributed TT-DNN with parallel paths. The TT-DNN provides a compact representation by decomposing each DNN layer into a TT format with fewer model parameters, and is well-designed to adapt to the block structure, pilot density, and the number of MIMO antennas. Moreover, the channel estimation is performed in block-by-block and antenna-by-antenna manners such that both input dimensions of TT-DNN and the number of model parameters can be further reduced. In addition, a preliminary stage of model pre-training is set for the DNN/TT-DNN channel estimator which boosts the channel estimation accuracy. Moreover, the proposed TT-DNN is generalized to semiblind channel estimation scenarios where there exists only preamble training symbols. Our experiments show that the proposed TT-DNN based channel estimator outperforms the DNN counterparts in terms of convergence rate, estimation accuracy, and robustness, and presents better performance than recurrent neural network for semiblind channel estimation.