The Movement for Change, Gorran, and the Patriotic Union of Kurdistan (PUK) said in separate statements several of their offices in the Duhok region, north of the Kurdish capital Erbil, were looted or burnt overnight.
No casualties were reported.
The semi-autonomous Kurdistan Regional Government (KRG) in northern Iraq said it had ordered the local police forces, known as Asayish, to stop the attacks. Barzani said on Sunday he would give up his position as president on Nov. 1 after an independence referendum he championed in northern Iraq backfired and triggered military and economic retaliation by the Iraqi government.
Armed protesters stormed parliament as it met on Sunday to approve his resignation. Opposition MPs who had been barricaded inside managed to leave later, according to their parties.
Gorran and the PUK both support Kurdish self-determination but Gorran opposed the referendum, saying it was ill-timed. The PUK supported the vote half-heartedly. In a televised speech announcing his plan to step down, Barzani said followers of rival PUK founder Jalal Talabani, who died in early October, had been guilty of "high treason" for handing over the oil city of Kirkuk to Iraqi forces without a fight two weeks ago.