Hate to be the one to say it... But this is asb. Persia is blocked off by two continents from America. They're based too far east to expand to a conveniant area for colonization of America. Persia also has a well established tendency to expand east, not west, primarily because that is where wealth was concentrated. The best I could see is a Persian cultured state in Morrocco maybe, but even that stretches disbelief, not to mention is against your op.
In reference to the idea of the earlier posters idea of a persianate Morocco, how about this. Absolute, absolute, Timur wank. Have him forget about the Delhi Sultanate, Russia and China and have him attempt to re-instate the Umayyad Caliphate, to the point he goes from Persia all the way to Morocco.
Also have him divide his empire amongst his generals and send a Persian to the Maghreb (maybe Timur or a successor also takes Granada/Andalusia). Maybe a few generations and successful management after this Persian ruled dynasty could established a few colonies.
Also to really tie it in to Persia let's have the actual Timurids dynasty be ruling in Persia, Transoxiana, Levant and Egypt so this Maghreb state could be seen as a client state of the former.
Would it, as a vassal state of the main Persian dynasty count? You could also have Persians filling the role that the Chinese did in South East Asia, as merchant settlers, in both the Maghreb and the new world.
What about having the Persian Shah somehow become fascinated by exploration during his visit to the court of French King Louis XIV and thus having him convince some European country to sell him a few tiny territories in the Americas for a large amount of cash?With a POD at really any point after the Fall of Rome, have a Persian state (based in their homeland, not a successor state with Persian culture somewhere more favorable) acquire colonies in the Americas. Bonus points if actually sizable Persian settlement occurs.