FORTUNE -- Dear Annie: I hope you and your readers have some suggestions for me, because I'm just about at the end of my rope. My 89-year-old mother, who has what her doctor calls "moderate" Alzheimer's disease, came to live with us a few months ago and needs constant supervision. We are lucky enough to have a licensed practical nurse who comes in on weekdays to be with her, but the nurse leaves at 3 p.m., which is right around the time my two teenaged kids get home from school. They've been great about pitching in, but I don't feel it's fair to ask them to give up extracurricular activities in order to keep an eye on my mom. The long and short of it is I really would like to be able to work from home in the late afternoons and early evenings.
The problem is my company expects everyone to be at his or her desk from 9 to 5 (or later), no exceptions. With the technology we have now, I could work more flexible hours without any problem, but my boss says there is no policy allowing this and that it would be "disruptive" to the office routine. How can I convince him otherwise? -Frazzled
Dear Frazzled: You surmise correctly that many other people share this dilemma, or a similar one. According to a raft of recent surveys, the so-called sandwich generation -- made up of people like you who are trying to care for children and parents at the same time -- is under more pressure than ever these days, partly due to ever-lengthening workdays brought on by the recession.
A whopping 89% of Americans say that balancing work and the rest of life is a problem, and more than a third (38%) say it has gotten worse because of the economic downturn, according to a new poll by research firm StrategyOne (www.strategyone.net).
More than 80% of Baby Boomers ages 45 to 54 are experiencing "high levels of stress" from juggling responsibilities at work and at home, says another survey, this one by the Hartford Financial Services Group and consulting firm ComPsych. Nearly half (46.6%) said that they were worried about how care giving is affecting their job performance.
The good news here is that many employers are aware of the struggle: About one-third of employees that responded to a recent global workforce study by the human resources consulting firm Towers Watson said that they were permitted to work from home either full-time or part-time, and an additional 50% said that they have the green light to do so "occasionally."
Moreover, new research by a nonprofit called WorldatWork suggests that -- partly in hopes of keeping their best people from quitting when hiring finally picks up again -- employers are showing more interest in offering their workers help with work-life balance, including flextime.
Elder care programs in particular are on the rise. Jamie Ladge, a management and organizational development professor at Northeastern University in Boston, notes that about 33% of large employers in the U.S. now offer elder care assistance of some kind, up from about 15% just 10 years ago.
Talkback: Does your company allow flextime? Leave your comments at the bottom of this story.
Against that backdrop, your company's rigid no-flextime policy seems out of step with the times. To change that, you'll need to present the idea in the same way you would put forward any other business proposal.
First, make the case that working flexible hours won't damage your productivity -- and may even improve it.
"Flextime employees are just as productive as other employees," says Ladge. "The problem is that some bosses don't believe it."
So start by marshalling some supporting evidence. The Towers Watson 2010 Global Workforce study mentioned above, for instance, says that people who work off-site some of the time are just as productive (41%) as their deskbound colleagues, or more so (47%). Only 11% of those 20,000 poll respondents said that flextime damages productivity. (1% had no opinion.)
You might also mention a report from the White House Forum on Workplace Flexibility, a gathering of businesspeople, policymakers, and labor leaders hosted by the president and Michelle Obama last spring. The benefits to companies of adopting flextime, the report says, "can outweigh the costs by reducing absenteeism, lowering turnover, improving the health of workers, and increasing productivity."
Then, focus on specifics. Make a list of your boss's likely concerns about how your absence from the office in the late afternoon will affect the rest of your team. For instance, let's say there's a meeting that is regularly scheduled for 4 p.m. Do you plan to be there virtually via Skype, by speaker phone, or by some other way? Your proposal should spell that out.
Likewise, if you have coworkers who need to speak with you on short notice, describe the various ways you would be reachable off-site -- by email, cell phone, and so on -- and note that you will give all the relevant contact information to everyone who might need it.
In short, the more details you can provide on how this would work, the more willing your boss may be to let you give it a try.
"Ask for a short trial period, say two weeks, or 30 days," Ladge suggests. "Then use that as an opportunity to prove you can work flexible hours and still get the job done."
Talkback: Does your company allow flextime? If so, does working flexible hours make you less productive, or more so? If you are a manager, what would persuade you to agree to let an employee work flexible hours? Tell us on Facebook, below.