Did Persian ever have a hard or soft "th" sound?

Farsi does not distinguish between ث (soft 'th' in Arabic, like "think") and ذ (hard 'th' in Arabic, like "that"). A native Farsi speaker pronounces ث like the 's' in "sing" and ذ like the 'z' in "zoo". For example, a Farsi speaker would pronounce Arabic loan words like ثریا and ذوالفقار as as "Soraya" and "Zolfaqar" respectively.

However, I've noticed some Persian words that are written using these consonants--like کیومرث ("Qumars", name) and گذشت ("gozasht", to pass)--but the words are pronounced normally (no 'th' sound).

Why, and what happened? Did Persian (Old, Middle, or New) at some point use 'th' sounds (phonemes?)? That would explain words like "Zarathustra" or "Azar".

Edit: Forgive me if I'm oversimplifying the label "Persian" for the language historically spoken in Iran. I'm an engineer by trade, so I'm not sure when to use Old/Middle/New Persian.