When Zhang Yiqiu found herself pregnant with her third child two years ago, the Beijing businesswoman expected to face a fine or bureaucratic hurdles. To her surprise, local officials praised her for her decision to have another child, she said, though they couldn’t provide the proper paperwork for her government health insurance to cover the birth because parents are still officially forbidden from having three children. Ms. Zhang isn’t typical in China, where the legacy of the one-child policy and the expense of raising children have meant many couples still have no more than one. However, interviews with parents, demographers and officials suggest that some local authorities are quietly allowing families to have a third child without the usual repercussions. China’s leaders, including President Xi Jinping, have said they are concerned about the country’s demographic trajectory. Abolishing the one-child policy in 2016 to let all couples have two children […]