The surge in violence by Afghan allies against their international partners has raised doubts about the ability of the two forces to work together at a key transition time. Afghan forces are expected to take over security for the country by the end of 2014, when the majority of international combat forces are scheduled to leave.
On the other side, a coalition airstrike killed dozens of Taliban militants, including one of their leaders, officials said.
Few details were immediately available about Sunday's killing of a coalition member in southern Afghanistan. NATO said only that they and Afghan authorities were investigating. Afghan officials could not be reached for comment.
The Taliban have been actively recruiting members of the Afghan security forces, saying in a statement last week that they considered these turncoat attacks a major part of their strategy against international forces.
Once an anomaly, these attacks have been climbing in recent months. There have been 30 such turncoat attacks so far this year, up from 11 in 2011.
On Saturday Defense Secretary Leon Panetta called Afghan President Hamid Karzai to encourage him to work with U.S. commanders to ensure more rigorous vetting of Afghan recruits.
On Friday it was disclosed that U.S. troops have been ordered to carry loaded weapons at all times in Afghanistan, even when they are on their bases. The order was a precaution against such insider attacks.
The monthlong Ramadan fasting period that ended Sunday has been particularly violent. The Eid al-Fitr holiday on Sunday continued the trend.
Insurgents killed two pairs of brothers with links to the government as well as three NATO service members in three separate attacks.
In the first attack, a bomb hidden in a cemetery in the southern province of Helmand killed a police chief and his brother who were visiting a family grave for the holiday.
Seven of the men's family members were wounded in the early-morning blast in the provincial capital of Lashkar Gah, said the Helmand deputy police chief, Ghulam Rabbani.
No one claimed responsibility, but the attack was consistent with the Taliban's strategy to target authorities and others who align themselves with the government or international forces.
The two men were brothers of a lawmaker for Helmand province, Abdulwadood Popal, who was not at the cemetery at the time of the blast. The family was visiting the grave after attending a morning prayer service for the holiday, which ends the month-long Ramadan fasting period.
Later in the western Farah province, gunmen on a motorcycle opened fire on the car of an intelligence service official as he was driving home from a family visit, killing him and his brother, who worked for the customs service.
Another relative was wounded, provincial deputy police chief Ghulam Ghows Malyar said.
In central Afghanistan, three NATO service members were killed when a vehicle struck a roadside bomb, officials said. Bamiyan Gov. Habiba Sarabi said the blast went off in Kohmard district while the troops were out on patrol.
NATO forces confirmed that three coalition service members were killed in a bombing, but did not provide their nationalities or other details.
Taliban spokesman Zabiullah Mujahid claimed responsibility for the attack in a statement.
Including Sunday's deaths, at least 41 international troops have been killed so far this month in Afghanistan.
In a speech marking the holiday, Karzai condemned the Ramadan attacks.
"The enemies of Muslims ... during the holy month of Ramadan treated the nation of Afghanistan cruelly: bombs, explosions in mosques, suicide attacks in mosques," Karzai said.
He challenged the Taliban to disavow them.
"If you are not behind this, it is being done in your name. As Muslims, as Afghans, raise your voice and say that you did not do it," he appealed.
In a message ahead of Eid al-Fitr, Taliban leader Mullah Mohammad Omar instructed his fighters once again to avoid killing or wounding Afghan civilians.
The Taliban have said previously, however, that they do not consider those who collaborate with the government to be civilians.
Killings of Afghan civilians have surged this year, according to the United Nations. Civilian deaths from targeted killings and assassinations jumped 34 percent for the first six months of 2012 to 255 killed, from 190 in 2011, the U.N. said in a report issued earlier this month.
Also Sunday, officials said that an airstrike by coalition forces in the northeast killed a large group of Taliban fighters and a local insurgent leader. At least two dozen insurgents were killed in the attack in Kunar province, said NATO forces spokesman Maj. Martyn Crighton.
Kunar Gov. Sayed Fazlullah Wahidi said the strike killed as many as 50 insurgents who had massed in a remote area of the province. He said it was not clear why they had gathered.