Eh, not really.

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The phrase you will come across in Herodotos and others is "The Persians worship the sky, which they refer to as Zeus". "Zeus" in this case is presumably Aúramazda, probablyt conflated with Elamite sky deity Humban. Anahita, a water goddess whose cult apparently merged Persian and Babylonian beliefs, was variously referred to as "Artemis" and "Aphrodite". But as you'll recognize, this is bog-standard Interpretatio Graecia. It is perhaps a little more interesting to note that later Parthians and Sasanian Persians produced some Greek parallel inscriptions, which still do use Zeus as an approximation of Aúramazda (or in Middle Persian, Ohrmazd).

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About the one decent Greek-language take on Persian beliefs occurs in Plutarch's Isis and Osiris, 46-47, which as you might recognize is... a little later than the era you'll asking about (viz., about half a milennium later). But it is also interpreting these beliefs through what seems to be a peculiarly late Hellenistic (Neoplatonist?) lens, not telling us a lot about earlier Greek understandings.

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If it lends any context, it is helpful to remember that Near Eastern mythos were also significantlly influenced by Indo-European beliefs, not least through the Hittites. Hence why we see e.g. the slaying of the serpent as a re-occuring motif in Canaanite and Mesopotamian religon.

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Yep, sorry to disappoint, would've been nice if it were the case. As a rule of thumb, in a mostly oral culture, 1000 years of separation is enough to forget just about any kinship or common origin,