Was the Chumash ever translated into Old Persian or Avestan? If not what about any other parts of the Tanach?
The manuscript ms. Or. 5446 in the British Library is the earliest dated Judeo-Persian document (6 March, 1319 CE) and it has a translation of the Chumash. There is one also in the Vatican Library from c. 15th century CE (Vat. Pers. 61) and another from the JTS (ms. L188 [Adler B.63]).
The translation of Ya`aqov ben Yosef Ṭāvus was published by Eliezer ben Gershom Soncino in 1546.
Fragmentary pre-Mongol tafsirs (generally from c. 11th century CE or after) exist as well.
Though none of the manuscripts above are in Old Persian, they are in early forms of Modern Persian.
For more on this, see "Judeo-Iranian Languages" by Habib Borjian in the Handbook of Jewish Languages.
I'm not sure if this is what you are looking for. It is a Chumash with Farsi translation.
And this link is for the book of Tehillim also in Farsi.
"Avestan" is the language used in the Avesta, the holy writings of the Zoroastrians and is the language originating from eastern Iran and western India. This religion, to my understanding, is forbidden for Jews.
Although it would seem unlikely that the Torah would have been translated into Avestan, it is possible that such a thing was done in the past for the Jewish community in Cochin or elsewhere.