SAN FRANCISCO — San Francisco became the first place in the country Tuesday to require businesses to provide fully paid leave for new parents in what was hailed as the latest move to address income inequality in the nation. The measure approved unanimously by the Board of Supervisors will give new mothers and fathers six weeks of fully paid time off, a rarity now offered to some government sector workers and some private employees, particularly those who work in the tech industry. “The vast majority of workers in this country have little or no access to paid parental leave, and that needs to change,” Supervisor Scott Wiener, who pushed the measure, said at a news conference before the vote. The measure requires another formal vote by the board next week as well as approval by Mayor Ed Lee, who says he is happy to sign it. Small business owners complained that the mandate is the latest in a long list of city mandates — including paid sick leave and health coverage — that unfairly targets them. “They don’t necessarily have the resources, they can’t absorb the increases in cost, and they feel like it’s kind of relentless, it’s one thing after the next,” said Dee Dee Workman, vice president of public policy at the San Francisco Chamber of Commerce. The issue of paid parental leave is gaining momentum across the country much like the debate over a higher minimum wage. Paid leave has become a topic in the presidential campaign as companies, especially in Silicon Valley, start offering better benefits.