The Persian Empire history is divided in 7 different stages that start in 550 BC and last until 1925, which is really impressive. During their first period from 550 BC until 330 BC, also known as the Achaemenid Empire, they conquered multiple nations and expanded their territory. They even defeated the Egypt and took control of their cities. They also fought against the Greek city-states (one of this battles is acted in the famous movie 300).
Thus, the right answer is : Persian Empire was in existence at the same time as (option D) ancient Egypt's Late Period.