The Jamaatul Ahrar group led by Omar Khorasani released the details of Hanifullah alias Hamza, claiming that he carried out the suicide bombing at the ceremony, a display of military pageantry which attracts thousands of spectators.
The outlawed group also released a photo of the bomber who belonged to a bordering town in Mohmand tribal agency.
The bomber blew himself up at the entrance of the parade complex soon after the completion of the ceremony last Sunday, said Ehsanullah Ehsan, a spokesman of the militant group.
The horrific attack at the entrance of the Wagah border parade venue killed 61 people, including 10 women and seven children, and left more than 110 injured.
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Officials said the damage could have been greater if the attacker had managed to penetrate into the main ceremony area.
Ehsan said they chose Wagah Border because of its "international significance" and that it is occupied by more security personnel. He warned such attacks would continue.
Another banned terrorist group Jandullah had also claimed responsibility of the attack.