Efficiently monitoring the condition of civil infrastructure requires automating the structural condition assessment in visual inspection. This paper proposes an Attention-Enhanced Co-Interactive Fusion Network (AECIF-Net) for automatic structural condition assessment in visual bridge inspection. AECIF-Net can simultaneously parse structural elements and segment surface defects on the elements in inspection images. It integrates two task-specific relearning subnets to extract task-specific features from an overall feature embedding. A co-interactive feature fusion module further captures the spatial correlation and facilitates information sharing between tasks. Experimental results demonstrate that the proposed AECIF-Net outperforms the current state-of-the-art approaches, achieving promising performance with 92.11% mIoU for element segmentation and 87.16% mIoU for corrosion segmentation on the test set of the new benchmark dataset Steel Bridge Condition Inspection Visual (SBCIV). An ablation study verifies the merits of the designs for AECIF-Net, and a case study demonstrates its capability to automate structural condition assessment.