How did Persia become involved in the Peloponnesian Wars?

Sparta had destroyed the Athenian navy, so Persia attacked Sparta.
The Persians had burned Athens, so Persia was attacked by Sparta.
Athens had defeated Persia at the Battle of Salamis, so Persia supported Sparta.
Sparta had resisted Persia at the Battle of Thermopylae, so Persia supported Athens.