Intervertebral disc disease, a prevalent ailment, frequently leads to intermittent or persistent low back pain, and diagnosing and assessing of this disease rely on accurate measurement of vertebral bone and intervertebral disc geometries from lumbar MR images. Deep neural network (DNN) models may assist clinicians with more efficient image segmentation of individual instances (discs and vertebrae) of the lumbar spine in an automated way, which is termed as instance image segmentation. In this work, we proposed SymTC, an innovative lumbar spine MR image segmentation model that combines the strengths of Transformer and Convolutional Neural Network (CNN). Specifically, we designed a parallel dual-path architecture to merge CNN layers and Transformer layers, and we integrated a novel position embedding into the self-attention module of Transformer, enhancing the utilization of positional information for more accurate segmentation. To further improve model performance, we introduced a new data synthesis technique to create synthetic yet realistic MR image dataset, named SSMSpine, which is made publicly available. We evaluated our SymTC and the other 16 representative image segmentation models on our private in-house dataset and public SSMSpine dataset, using two metrics, Dice Similarity Coefficient and the 95th percentile Hausdorff Distance. The results indicate that SymTC surpasses the other 16 methods, achieving the highest dice score of 96.169 % for segmenting vertebral bones and intervertebral discs on the SSMSpine dataset. The SymTC code and SSMSpine dataset are publicly available at https://github.com/jiasongchen/SymTC.