The Persians relates to modern day because it focuses centrally on a devastating military defeat brought about by the arrogance and overreach of a ruler who thinks he cannot lose. In this case, it is Xerxes taking on the Greeks, going so far in his confidence of winning as to build bridges that the Greek can later use against him to invade from other directions.
The parallels today to Xerxes's military overreach are not hard to see. Powerful nations that have not known defeat in living memory tend to believe they are invincible. This can lead to them getting involved in unnecessary and disastrous wars. One modern example in the Boer War, which so depleted the British treasury that the country was weakened going into World War I. World War I was another case of a disastrous and unnecessary war. In more recent times, the US has gotten involved in vastly expensive wars in the Middle East that have arguably done more harm than good, because our leaders have had feelings of invincibility just as Xerxes did.
While the modern US does not have to fear total devastation from its wars in the Middle East, as The Persians emphasizes, the nation as a whole suffers from wars entered into thoughtlessly and from arrogance. Today, we all have to pay the tax bill for Middle East wars, and we all have to deal with the death, disability, and trauma of loved ones and returning troops, not to mention the horror we might feel at the suffering and devastation unleashed on innocent people.