Over the past few years, tasks such as automatic information extraction and object surface detection from 3D medical images have become increasingly popular, providing clinicians with numerous new opportunities for exploration and diagnostic assistance. Nonetheless, the segmentation process can present various challenges. This paper introduces an automatic 3D deep parametric active surface model (3D B-snake) approximated by bivariate Unified and Extended spline (UE-spline) functions with a parameter controlling the shape of the snake. We introduce an energy term that enables to separate the different textures present in an image using a Convolutional Neural Network (CNN) learning model, namely U-Net. After balancing our snake surface to align it with the complex parts of the object to be segmented, an object snake alignment strategy is proposed using an interpolation scheme that exploits some geometric properties of the object. Experiments performed on the BraTS21 dataset show that our model is very powerful to deal with various difficulties encountered in medical image segmentation.