CHENGDU, China — In response to inquiries from grieving relatives, local officials announced Monday that parents whose only child was killed or grievously injured in the May 12 earthquake would be exempt from the country’s one-child policy.

The exception, issued by the Chengdu Population and Family Planning Committee in Sichuan Province, said qualified parents could apply for legal permission to have another child, according to The Associated Press.

Thousands of parents have openly challenged the government over why so many schools collapsed during the earthquake. An estimated 10,000 students are believed to have died.

The anguish of parents and grandparents has been compounded by the one-child policy, which was introduced in 1979 to control population growth. Provincial officials, especially those in rural areas or in regions with large minority populations, are sometimes given latitude in the application of the regulations. In some places, for example, families are permitted to have more than one child if the first is a girl.

According to the policy, local governments can levy steep fines on couples who have more than one child; the children of those who defy the rules are sometimes denied government benefits, including access to a free education.

The committee announced Monday that if a couple’s legally born child was killed in the earthquake, an illegal child under 18 years could be registered as a legal replacement. If the dead child was illegal, it said the family would no longer be responsible for outstanding fines, although parents would not be reimbursed for penalties already paid.

The changes, however, may come as little solace to parents who have only a photo, a backpack or the ashes of their dead son or daughter. Zhongxin Sun, a sociology professor at Fudan University in Shanghai, said some mothers may be too old to conceive; others may have undergone sterilization. “To lose a child is to lose everything for Chinese parents,” said Professor Sun, who is a visiting scholar at Yale University Law School. “A child is their only hope.”