Automatic Modulation Recognition (AMR) plays a critical role in wireless communication and can be applied in various applications such as spectrum monitoring and signal surveillance. Recently, different AMR approaches have been proposed based on Deep Learning (DL), which are inspired by the powerful abilities of feature extraction in neural networks, i.e., Convolutional Neural Networks (CNN) and Recurrent Neural Networks (RNN). Among these solutions, hybrid methods extract spatial features and temporal features of signals by combining CNN and RNN into a single architecture to provide better performance. However, it is inefficient to extract two types of features using a single architecture since the temporal feature could be degraded after CNN layers, and the spatial feature may be compromised after RNN layers. To overcome this challenge, we propose a parallel neural network architecture for AMR, which extracts spatial features using CNN layers and temporal features using LSTM (Long Short-Term Memory) layers, respectively, in two parallel routes. Afterward, the extracted features will be combined to predict the modulation scheme of a signal. Extensive simulations are conducted on signals with 11 different modulation methods at a wide range of Signal-to-Noise Ratio (SNR) levels to evaluate our proposed parallel feature extraction architecture. In addition, we compare our solution against three other DL-based methods, and the results show that our method outperforms in terms of recognition accuracy.