Persia would have to solidify itself before it could really do any annexing. Persia was divided into northern and southern spheres of influence under the
Anglo-Russian Treaty of 1907, with the Russians having the North and the British the South. So, you'd probably need some event around 1910-1913 where Persia could centralize into a more coherent state. Perhaps Persia strengthens itself and is able to reduce Anglo-Russian oversight in exchange for treaties guaranteeing a Persian counterweight to the Ottomans.
Assuming all that happens, then a Collapse of the Ottomans might allow the Persians to seize parts of what is now Iraq. Perhaps this implies somewhat more of the planned partitioning under the Treaty of Sevres is carried out, but I can't see the W. Allies exerting the effort to make sure it really happens. In reality, the most that's likely is Greece getting the Western side of the Bosphorus and Armenia, Kurdistan, Jordan, Syria, etc being made independent. However, even then, Turkey would likely try to take over Armenia (as in IOTL) and possibly Kurdistan and Russia (as IOTL) would seize Armenia.
So, Turkey would probably default to nearly what it is now. Persia might only gain small chunks of Eastern Iraq.