Maintaining and monitoring civil infrastructure is crucial for ensuring safety and functionality, as these structures are vital for transportation and economic development. Environmental exposure and increasing traffic loads degrade structural integrity over time, raising the risk of damage. Corrective and preventive maintenance, including regular inspections and monitoring, are essential to mitigate these risks. Static load tests, commonly used to evaluate bridge behavior, are resource-intensive, posing challenges for nations with extensive infrastructure networks. This study proposes a cost-effective method using unmanned aerial vehicles (UAVs) to measure vertical displacements in riveted steel bridges. By leveraging aerial photogrammetry, digital geometrical models of the structure are created at different loading stages. Machine learning algorithms identify and track rivets, comparing their positions before and after loading to estimate deflections. Tested on a real-world bridge in Toledo, Spain, the method achieved millimetric precision, matching the performance of commercial off-the-shelf potentiometers. This UAV-based approach streamlines infrastructure assessments, reduces costs, and enhances safety, offering a scalable solution for aging structures.