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This article presents Holistically-Attracted Wireframe Parsing (HAWP), a method for geometric analysis of 2D images containing wireframes formed by line segments and junctions. HAWP utilizes a parsimonious Holistic Attraction (HAT) field representation that encodes line segments using a closed-form 4D geometric vector field. The proposed HAWP consists of three sequential components empowered by end-to-end and HAT-driven designs: (1) generating a dense set of line segments from HAT fields and endpoint proposals from heatmaps, (2) binding the dense line segments to sparse endpoint proposals to produce initial wireframes, and (3) filtering false positive proposals through a novel endpoint-decoupled line-of-interest aligning (EPD LOIAlign) module that captures the co-occurrence between endpoint proposals and HAT fields for better verification. Thanks to our novel designs, HAWPv2 shows strong performance in fully supervised learning, while HAWPv3 excels in self-supervised learning, achieving superior repeatability scores and efficient training (24 GPU hours on a single GPU). Furthermore, HAWPv3 exhibits a promising potential for wireframe parsing in out-of-distribution images without providing ground truth labels of wireframes.