7T MRI scanner provides MR images with higher resolution and better contrast than 3T MR scanners. This helps many medical analysis tasks, including tissue segmentation. However, currently there is a very limited number of 7T MRI scanners worldwide. This motivates us to propose a novel image post-processing framework that can jointly generate high-resolution 7T-like images and their corresponding high-quality 7T-like tissue segmentation maps, solely from the routine 3T MR images. Our proposed framework comprises two parallel components, namely (1) reconstruction and (2) segmentation. The reconstruction component includes the multi-step cascaded convolutional neural networks (CNNs) that map the input 3T MR image to a 7T-like MR image, in terms of both resolution and contrast. Similarly, the segmentation component involves another paralleled cascaded CNNs, with a different architecture, to generate high-quality segmentation maps. These cascaded feedbacks between the two designed paralleled CNNs allow both tasks to mutually benefit from each another when learning the respective reconstruction and segmentation mappings. For evaluation, we have tested our framework on 15 subjects (with paired 3T and 7T images) using a leave-one-out cross-validation. The experimental results show that our estimated 7T-like images have richer anatomical details and better segmentation results, compared to the 3T MRI. Furthermore, our method also achieved better results in both reconstruction and segmentation tasks, compared to the state-of-the-art methods.