The utilization of different types of brain images has been expanding, which makes manually examining each image a labor-intensive task. This study introduces a brain tumor segmentation method that uses two parallel U-Net with an asymmetric residual-based deep convolutional neural network (TPUAR-Net). The proposed method is customized to segment high and low grade glioblastomas identified from magnetic resonance imaging (MRI) data. Varieties of these tumors can appear anywhere in the brain and may have practically any shape, contrast, or size. Thus, this study used deep learning techniques based on adaptive, high-efficiency neural networks in the proposed model structure. In this paper, several high-performance models based on convolutional neural networks (CNNs) have been examined. The proposed TPUAR-Net capitalizes on different levels of global and local features in the upper and lower paths of the proposed model structure. In addition, the proposed method is configured to use the skip connection between layers and residual units to accelerate the training and testing processes. The TPUAR-Net model provides promising segmentation accuracy using MRI images from the BRATS 2017 database, while its parallelized architecture considerably improves the execution speed. The results obtained in terms of Dice, sensitivity, and specificity metrics demonstrate that TPUAR-Net outperforms other methods and achieves the state-of-the-art performance for brain tumor segmentation.