The KDP actually invited Saddam's army into Kurdistan to help it smash the PUK --- over 30,000 Iraqi troops took Irbil from Talabani's PUK and handed it over to the PUK. The KDP also was helped by Iraqi Army troops when it took Sulamaniyah.

Talabani's PUK is quite tight with the Iranians going back to the Iran-Iraq War (Saddam had similar warm relations with Iranian Kurds that fought for, yes, Iraq during that conflict). Barzani's KDP has also at times sought refuge behind the Iranian border.