We introduce NeuBase , a neural parametric surface representation that both accurately fits target surfaces with fine geometric detail and supports intuitive real time surface deformation. NeuBase consists of a Catmull-Clark subdivision base surface and an offset field defined by a set of neural basis functions encoded via a neural map. By construction, NeuBase surfaces exhibit four fundamental geometric properties, i.e., linearity, locality, smoothness, and affine equivariance, enabling real-time, direct manipulation without retraining the neural network. In addition, we propose a scalable neural map that maintains memory efficiency even for complex shapes with dense control meshes. Experiments on a large-scale dataset demonstrate that our method achieves better fitting accuracy than state-of-the-art neural parametric surface representations.