The militant group also chastised the Afghan government for delaying the release of 5,000 Taliban prisoners promised in the agreement, AP reported.
Kabul was not a party to the negotiations that led to the deal signed in February.
Taliban negotiators signed the deal with the US in Qatar on February 29. Under the deal, the US committed to a gradual withdrawal of its forces from Afghanistan, and the Taliban agreed not to allow terrorist groups on Afghan soil and not to attack US and international forces as long as they were in the process of withdrawing.
The US and other foreign forces have started pulling out, a process that could take 14 months to complete, Press TV reported.
The Taliban would refuse at the time to recognize the Afghan government. But under the same deal, the group agreed to ultimately sit down with the government. A first face-to-face meeting over a prisoner exchange did take place last week.
The Taliban said in the Sunday statement that they had restricted attacks against Afghan security forces to rural outposts, had not attacked international forces and had not attacked Afghan forces in cities or military installations, AP reported.
But the Taliban have not stopped attacking government forces. Since the deal was signed, Taliban assaults have claimed the lives of scores of security forces.
The Taliban said these limits on their attacks had not been specifically laid out in the agreement with the US signed in February.
The Taliban warned of more violence if the US and the Afghan government continue alleged violations of the deal.
The militant group threatened that it would “increase the level of fighting” if the alleged violations continued.
“We are seriously asking the Americans to abide by the contents of the agreement and to alert their allies to fully abide by the agreement,” the Taliban statement said.
A spokesman for the US military in Afghanistan in a tweet rejected the Taliban statement, claiming the group’s comments were “baseless.”
There was no immediate reaction from the Afghan government, Press TV reported.
The US and NATO have already begun to withdraw troops from Afghanistan. The full withdrawal is expected to be completed in 14 months and is tied to Taliban commitments to fight terrorist groups and help in the battle against the Daesh.
The withdrawal is not tied to the success of intra-Afghan negotiations.
In the Afghan capital, President Ashraf Ghani announced his new cabinet as he squabbles with his main political challenger over last year’s election results. Ghani’s move came even as Afghan mediators — including former president Hamid Karzai — shuttled between the president and his opponent, Chief Executive Abdullah Abdullah, who has also declared himself Afghanistan’s president.
US Secretary of State Michael R. Pompeo had traveled to Afghanistan last month to try to break the impasse between Ghani and Abdullah. Pompeo left without a solution; however, last week he welcomed that the Afghan government had put together a negotiating team and made progress toward the prisoner releases.
Those releases have stumbled even as the Taliban sent a three-member team to Kabul last week.