Avi K, of course Yemenites can speak Persian, the question is if they are indeed speaking or spoke that language. Persian is not Arabic at all and is completely different. No one speaking Arabic would say they are speaking Persian and vice versa. Perhaps the people who told me they were speaking Persian between themselves were wary of saying they were speaking Arabic. I’m not saying Wikipedia is the authority in everything and everyone but someone who is familiar with the subject wrote on the page of Jewish Yemenites that they speak Judeo-Yemeni Arabic and Arabic is different than Persian and if there are any similarities at all between the two languages, certainly the Judeo version of Arabic is an even greater difference from the Persian language.
The descendents of Mizrachi Jews in Israel have indeed mostly adopted the Israeli accent which is Sephardi. And that’s why I was confused where this accent is from. After you told me that Persian Jews pronounce the “hes” I’ve done more research which and learnt that many older Mizrachis in Israel and some Israeli singer retained the “hes” but most of the rest of Mizrachi dropped the Mizrachi Jewish dialect and are speaking with the Israeli Hebrew dialect. That’s why I was confused where that dialect is from since some Mizrachis still spoke with that accent and some didn’t.
Ashkenazim have a soft and hard ת.
Regarding some communities pronouncing the “ד” with the “th” sound, I have read an article where a Yemenite said that their pronunciation of the LH letters were the original pronunciations and he gave an example that according to halacha the letter “ד” from the word “אחד” in Krias Shma needs to be drawn out but that is only possible if it is pronounced with the “the” sound. ( I tried it but it’s hard for me to do, firstly because Ashkenazim were taught to draw out the ח and secondly the “the” sound is not easy to stretch out but still doable as opposed to the hard “d” sound)