Pedestrian trajectory prediction is an important task in practical applications such as automatic driving and surveillance systems. It is challenging to effectively model social interactions among pedestrians and capture temporal dependencies. Previous methods typically emphasized social interactions among pedestrians but ignored the temporal consistency of predictions and suffered from superfluous interactions by dense undirected graphs, resulting in a considerable deviance from reality. In addition, autoregressive approaches predicted future locations conditioning on previous predictions one by one, which would lead to error accumulation and time consuming. To address these issues, we present Non-autoregressive Sparse Transformer (NaST) networks for pedestrian trajectory prediction. Specifically, NaST models sparse spatial interactions and sparse temporal dependency via a sparse spatial transformer and a sparse temporal transformer separately. Different from previous predictions such as RNN-based approaches, the transformer decoder works in non-autoregressive pattern and predicts all the future locations at one time from a query sequence, which could avoid the error accumulation and be less computationally intensive. We evaluate our proposed method on the ETH and UCY datasets, and the experimental results show our method outperforms comparative state-of-the-art methods.