This paper presents a recursive Bayesian inference framework for joint parameter-input identification, and virtual sensing for strain time history prediction of an offshore platform using sparse output-only measurements. The studied offshore platform, known as FINO3, is in the North Sea and is instrumented with a variety of sensors, including accelerometers and strain gauges. Offshore platforms are fatigue critical structures due to harsh marine environmental conditions and continuous cyclic wind and wave loads. Therefore, continuous monitoring of strain time histories at hotspot locations of offshore structures is important for reducing maintenance cost and avoiding unexpected failures. A windowed unscented Kalman filter (UKF) is employed to estimate an uncertain modeling parameter (foundation stiffness) and unknown input load time histories using output-only acceleration and strain measurements. The input loads are divided into overlapping windows, and windowed inputs and model parameters are combined as an augmented state vector in the UKF framework. Then strain time histories at critical locations are estimated through a virtual sensing strategy using the estimated input loads and model parameter. A traditional modal expansion approach combined with model updating is also implemented for the purpose of verification and comparison. The proposed method is first demonstrated through a numerical study using a finite element model of FINO3, where accurate model parameter and input estimations are obtained. Then the approach is further investigated using the actual measurements on FINO3. More accurate strain predictions are provided by the UKF than the modal expansion approach, which recommends the proposed UKF method for fatigue monitoring and input estimation.