Direct proof of use
The BMP-1 is directly documented in the 2003 Iraq War through official Marine Corps history and U.S. Army public-domain imagery. Marine Corps history of the 1st Marine Division's 2003 campaign records Iraqi armored vehicles at Basrah International Airport and states that Lance Corporal Adam Seeley destroyed a BMP-1 on the airfield during RCT-7's attack.
Later Operation Iraqi Freedom imagery separately documents post-invasion Iraqi security-force operation of the same vehicle family. A U.S. Army image hosted by Wikimedia Commons identifies Iraqi soldiers passing in a BMP-1 amphibious tracked infantry fighting vehicle in Sabe al Boor, Iraq, on December 21, 2006, during an Operation Iraqi Freedom mission.
Sources: With the 1st Marine Division in Iraq, 2003, BMP-1 Iraq 3 - Wikimedia Commons
Timeline
On March 21, 2003, Marine RCT-7 attacked Basrah International Airport during the opening ground campaign. The Marine Corps account describes Kilo Company moving through the airfield fence, fighting to support India Company's breach, and destroying armored vehicles, including a BMP-1.
On December 21, 2006, U.S. Army imagery showed Iraqi soldiers operating a BMP-1 in Sabe al Boor during Operation Iraqi Freedom. That image places the vehicle in Iraqi government and security-force use after the fall of Saddam Hussein's regular army.
Sources: With the 1st Marine Division in Iraq, 2003, BMP-1 Iraq 3 - Wikimedia Commons
Operational role
The sourced record places the BMP-1 on the Iraqi government and security-forces side of the conflict. In the invasion phase, the Marine Corps history documents the vehicle as Iraqi armored equipment encountered by U.S. Marines during the fight for Basrah International Airport, alongside other mechanized vehicles and infantry.
The 2006 image documents a different phase of the war: Iraqi soldiers operating a BMP-1 during Operation Iraqi Freedom. It supports continued use as an armored troop carrier and patrol platform by Iraqi forces rather than a coalition transfer claim by itself.
The available direct sources support battlefield presence, destruction of at least one Iraqi BMP-1 in March 2003, and later Iraqi Army operation in December 2006. They do not establish a complete order of battle, total quantities in service, or a continuous usage chronology across every phase of the war.
Sources: With the 1st Marine Division in Iraq, 2003, BMP-1 Iraq 3 - Wikimedia Commons