Machine scheduling in modern industries is a critical combinatorial optimization challenge for efficient resource allocation and decision-making. This problem involves assigning tasks to machines over a finite planning horizon, aiming to optimize scheduling criteria, such as completion time and makespan. Although theoretical advancements have been made in recent decades, practical applications often face uncertainties due to machine breakdowns and unpredictable job release dates, which require robust scheduling strategies. This research makes three main contributions to address these complexities. The first contribution is developing a mathematical framework that extends exact modeling approaches to parallel machine scheduling problems, adapting the bucket-indexed formulation to improve computational efficiency. This formulation accommodates realistic constraints, such as machine eligibility and task precedence, which are crucial for real-world applications. The second contribution focuses on applying robust optimization techniques to handle uncertainties in scheduling parameters. Building on recent advancements in the field, we propose adapting the bucket-indexed formulation to incorporate uncertain processing times within predefined uncertainty sets. This robust approach ensures resilient scheduling, which can adjust to uncertain scenarios and balance expected performance with the necessary robustness for different conditions. The third contribution addresses a recent and innovative issue: datadriven uncertainty representation. We conduct a case study involving offshore drilling rig scheduling in the oil and gas industry. This problem is formulated as a parallel machine scheduling problem, where uncertainties, such as variations in task processing times, play an important role. We use historical data to design a data-driven ellipsoidal uncertainty set, capturing the uncertainties in task processing times. The effectiveness of the proposed methodologies was validated through computational experiments conducted on various size-instances problems, ranging from small to large-scale. The results show that our contributions enhance parallel machine scheduling methodologies and have practical relevance, mainly when applied to real-world problems, such as offshore drilling rig scheduling, under deterministic and uncertain conditions, highlighting the importance of considering robustness in solutions for real-world industrial problems.