Accurate and timely mapping of crop types and having reliable information about the cultivation pattern/area play a key role in various applications, including food security and sustainable agriculture management. Remote sensing (RS) has extensively been employed for crop type classification. However, accurate mapping of crop types and extents is still a challenge, especially using traditional machine learning methods. Therefore, in this study, a novel framework based on a deep convolutional neural network (CNN) and a dual attention module (DAM) and using Sentinel-2 time-series datasets was proposed to classify crops. A new DAM was implemented to extract informative deep features by taking advantage of both spectral and spatial characteristics of Sentinel-2 datasets. The spectral and spatial attention modules (AMs) were respectively applied to investigate the behavior of crops during the growing season and their neighborhood properties (e.g., textural characteristics and spatial relation to surrounding crops). The proposed network contained two streams: (1) convolution blocks for deep feature extraction and (2) several DAMs, which were employed after each convolution block. The first stream included three multi-scale residual convolution blocks, where the spectral attention blocks were mainly applied to extract deep spectral features. The second stream was built using four multi-scale convolution blocks with a spatial AM. In this study, over 200,000 samples from six different crop types (i.e., alfalfa, broad bean, wheat, barley, canola, and garden) and three non-crop classes (i.e., built-up, barren, and water) were collected to train and validate the proposed framework. The results demonstrated that the proposed method achieved high overall accuracy and a Kappa coefficient of 98.54% and 0.981, respectively. It also outperformed other state-of-the-art classification methods, including RF, XGBOOST, R-CNN, 2D-CNN, 3D-CNN, and CBAM, indicating its high potential to discriminate different crop types.