The emerging field of combining compressed sensing (CS) and three-dimensional synthetic aperture radar (3D SAR) imaging has shown significant potential to reduce sampling rate and improve image quality. However, the conventional CS-driven algorithms are always limited by huge computational costs and non-trivial tuning of parameters. In this article, to address this problem, we propose a two-path iterative framework dubbed TPSSI-Net for 3D SAR sparse imaging. By mapping the AMP into a layer-fixed deep neural network, each layer of TPSSI-Net consists of four modules in cascade corresponding to four steps of the AMP optimization. Differently, the Onsager terms in TPSSI-Net are modified to be differentiable and scaled by learnable coefficients. Rather than manually choosing a sparsifying basis, a two-path convolutional neural network (CNN) is developed and embedded in TPSSI-Net for nonlinear sparse representation in the complex-valued domain. All parameters are layer-varied and optimized by end-to-end training based on a channel-wise loss function, bounding both symmetry constraint and measurement fidelity. Finally, extensive SAR imaging experiments, including simulations and real-measured tests, demonstrate the effectiveness and high efficiency of the proposed TPSSI-Net.