What role did Sparta and Athens play in the Persian War?

It was not a case of Sparta helping Athens, it was a case of mutual support. Persia was intent on establishing an ethnic frontier, as the mainland Greeks were helping rebellions in the Greek cities under Persian control in Asia Minor. So Persia decided to capture mainland Greece and end the problem.

The mainland Greek states met and decided on a mutual defence policy. Sparta was elected to be leader. Athens' major contribution was its navy, however Sparta commanded on land and sea.

Incidentally, several of the Greek states went over to Persia, whose agents had descended in advance on the nothern cities with lots of gold bribes, and promises to make particular factions the rulers of their cities under a Persian provincial governor.
They both joined forces to proctet themselfs from the Greeks