Final answer:
The Sassanid Persians, under King Shapur I, captured the Roman Emperor Valerian in 260 CE. This event signifies the hostile relations between the Sassanid Empire and the Roman Empire, with the former posing a significant threat to Rome's eastern provinces.
Step-by-step explanation:
The question concerns the Sassanid Persians and their interaction with the Roman Empire. Among the events that took place, the capture of the Roman emperor Valerian by the Persian king Shapur I is notable. This occurred in 260 CE when Valerian was leading a Roman army against Persia. The Sassanid Empire was founded in 224 by Ardashir and was characterized by continuous conflicts with the Romans, amongst others. By the fifth century, after a series of weak kings, priests largely controlled the administration of the empire.
Thus, the correct answer to the student's question is that the Sassanid Persians overran the eastern provinces and captured Emperor Valerian (option D). This reflects the turbulent relationship between the Romans and the Sassanids, including periods of warfare and power struggles that exemplify the regional dynamics of the time.