Real-life parallel machine scheduling problems can be characterized by: (i) limited information about the exact task duration at scheduling time, and (ii) an opportunity to reschedule the remaining tasks each time a task processing is completed and a machine becomes idle. Robust optimization is the natural methodology to cope with the first characteristic of duration uncertainty, yet the existing literature on robust scheduling does not explicitly consider the second characteristic - the possibility to adjust decisions as more information about the tasks' duration becomes available, despite that re-optimizing the schedule every time new information emerges is standard practice. In this paper, we develop a scheduling approach that takes into account, at the beginning of the planning horizon, the possibility that scheduling decisions can be adjusted. We demonstrate that the suggested approach can lead to better here-and-now decisions and better makespan guarantees. To that end, we develop the first mixed integer linear programming model for adjustable robust scheduling, and a scalable two-stage approximation heuristic, where we minimize the worst-case makespan. Using this model, we show via a numerical study that adjustable scheduling leads to solutions with better and more stable makespan realizations compared to static approaches.