Meanwhile, there was no change among mothers. That's at least partially because paid leave policies were already more widely used by women than men, according to the study. But the findings also unearthed a widespread lack of understanding about maternity leave benefits, particularly among lower-income mothers — less than 2% of whom had accurate information about the new policy.

Goodman called those findings "surprising and distressing," noting that lower-income workers were much less likely to say they understood their benefits and that their employers do anything to help, compared to those with a higher income.

"That's potentially disturbing," Goodman said. "This is a space where employers should really step in and provide more information and more support for workers who need to take leave."

One of the key takeaways, according to Goodman, is that without a one-size-fits-all approach to paid parental leave, we're left with individual policies that are too complicated.

"There's a lot of confusion all round," Goodman said. "To be more effective we should try to advocate for more simple universal policies, as opposed to right now we have this patchwork of policies at the city and state level, and they don't all align. So even if HR leaders really understood this and were motivated to talk to their employees about it, it's hard."

San Francisco's policy also doesn't cover small employers, meaning many low-income parents were left out.

But there was one source of information that both low and high-income employees consistently said they relied on: healthcare providers.

"I think there's some potential to work with healthcare providers or health systems to try to get some information out to workers that way," Goodman said.

Studies have linked paid parental leave to productivity and retention in the workplace and improved physical and mental health among workers. Goodman says that mirrors findings from a similar survey asking employers how they feel about San Francisco's new policy, publishing later this month.

"It seems like even with a policy like this that's relatively burdensome on employers because it asks employers to provide some pay, they were still supportive and said it didn't have a negative impact on them."

The best thing HR leaders can do, in Goodman's view, is support paid leave policies when they come up at the state and federal level, and to encourage the business community to speak up.

"From the business community, to have their support is so powerful," Goodman said. "In each of the states that have passed a paid leave policy, it's been critical to have businesses come onboard and say, 'Yes, we think this is an important benefit.' "

Read more:

• Parental leave is booming, just not for everyone
• More companies embracing gender-neutral parental leave policies
• Report weighs different paid leave plans