I'm not sure either when they immigrated to Siam, but after the fall of Ayutthaya some of the high ranking officials of the new Siamese kingdom, Rattanakosin, were of Persian decents...according to historical documents I've read online. There has always been migrations from other parts of the world since Ayutthaya era...it traded heavily with foriegners, and a lot of them decided to stay. A lot of Chinese, in particular, settle down in Ayutthaya. Even the today's king's family has some Chinese ancestries. In ceremonial events, for example, the Brahmans would be the ones who carried it out, seeing things through, these people are decendants from people, who migrated from Ceylon, I believe, centuries earlier. Siam was, and still is, a melting pot of cultures and idea. I went to school with a lot of Sikhs and Hindus in Thailand, when I was in 8th grade. There were also migration from Laos and Cambodia. Did you know there are more Laotions in Thailand than in Laos? And you probably know about the Shans already, they are exactly the same as people from Northern Thailand, but their kingdom is in Burma. They have their own distinct wirtting however, and it used to be that the Shan State was the biggest Thai kingdom, even bigger than Ayutthaya, but because of centuries of bloody warfare with the Burmans, it had reduced their number greatly. The Shans (and the Siamese, also) called themselves Thai Yai, which tranlates to Big Thai...because they were the largest Thai state in ancient time. I hope this clears things up for you.
