Extracting parametric edge curves from point clouds is a fundamental problem in 3D vision and geometry processing. Existing approaches mainly rely on keypoint detection, a challenging procedure that tends to generate noisy output, making the subsequent edge extraction error-prone. To address this issue, we propose to directly detect structured edges to circumvent the limitations of the previous point-wise methods. We achieve this goal by presenting NerVE, a novel neural volumetric edge representation that can be easily learned through a volumetric learning framework. NerVE can be seamlessly converted to a versatile piece-wise linear (PWL) curve representation, enabling a unified strategy for learning all types of free-form curves. Furthermore, as NerVE encodes rich structural information, we show that edge extraction based on NerVE can be reduced to a simple graph search problem. After converting NerVE to the PWL representation, parametric curves can be obtained via off-the-shelf spline fitting algorithms. We evaluate our method on the challenging ABC dataset. We show that a simple network based on NerVE can already outperform the previous state-of-the-art methods by a great margin.