Although most brain tumor diagnosis studies have focused on tumor segmentation and localization operations, few studies have focused on tumor detection as a time- and effort-saving process. This study introduces a new network structure for accurate glioma tumor detection and classification using two parallel deep convolutional neural networks (PDCNNs). The proposed structure is designed to identify the presence and absence of a brain tumor in MRI images and classify the type of tumor images as high-grade gliomas (HGGs, i.e., glioblastomas) or low-grade gliomas (LGGs). The introduced PDCNNs structure takes advantage of both global and local features extracted from the two parallel stages. The proposed structure is not only accurate but also efficient, as the convolutional layers are more accurate because they learn spatial features, and they are efficient in the testing phase since they reduce the number of weights, which reduces the memory usage and runtime. Simulation experiments were accomplished using an MRI dataset extracted from the BraTS 2017 database. The obtained results show that the proposed parallel network structure outperforms other detection and classification methods in the literature.