By WINNIE HUNOV. 1, 2015

Forty strollers were double- and triple-parked on the main floor of the Fort Washington Library in Upper Manhattan. As another one came through the door, Velda Asbury waved toward a spot beside a book stack.

Officially, Ms. Asbury is a library clerk, checking books in and out. But every Wednesday she doubles as a parking attendant during one of the New York Public Library’s most popular programs: story time.

Among parents of the under-5 set, spots for story time have become as coveted as seats for a hot Broadway show like “Hamilton.” Lines stretch down the block at some branches, with tickets given out on a first-come-first-served basis because there is not enough room to accommodate all of the children who show up.

Workers at the 67th Street Library on the Upper East Side of Manhattan turn away at least 10 people from every reading. They have been so overwhelmed by the rush at story time — held in the branch’s largest room, on the third floor — that once the space is full, they close the door and shut down the elevator. “It is so crowded and so popular, it’s insane,” Jacqueline Schector, a librarian, said.

Story time is drawing capacity crowds at public libraries across New York and across the country at a time when, more than ever, educators are emphasizing the importance of early literacy in preparing children for school and for developing critical thinking skills. The demand crosses economic lines, with parents at all income levels vying to get in.

Many libraries have refashioned the traditional readings to include enrichment activities such as counting numbers and naming colors, as well as music and dance. And many parents have made story time a fixture in their family routines alongside school pickups and playground outings — and, for those who employ nannies, a nonnegotiable requirement of the job.

In New York, demand for story time has surged across the city’s three library systems — the New York Public Library, the Brooklyn Public Library, and the Queens Library — and has posed logistical challenges for some branches, particularly those in small or cramped buildings. Citywide, story time attendance rose to 510,367 people in fiscal year 2015, up nearly 28 percent from 399,751 in fiscal 2013.

“The secret’s out,” said Lucy Yates, 44, an opera coach with two sons who goes to story time at the Fort Washington Library every week.

Stroller-pushing parents and nannies begin to line up for story time outside some branches an hour before doors open. To prevent overcrowding, tickets are given out at the New Amsterdam and Webster branches, both in Manhattan, the Parkchester branch in the Bronx, and a half-dozen branches in Brooklyn, including in Park Slope, Kensington and Bay Ridge.

The 67th Street branch keeps adding story times — there are now six a week — and holds sessions outdoors in the summer, when crowds can swell to 200 people.

In Queens, 41 library branches are scheduled to add weekend hours this month, and many will undoubtedly include weekend story times. As Joanne King, a spokeswoman for the library explained, parents have been begging for them and “every story time is full, every time we have one.”

Long a library staple, story time has typically been an informal reading to a small group of boys and girls sitting in a circle. Today’s story times involve carefully planned lessons by specially trained librarians that emphasize education as much as entertainment, and often include suggestions for parents and caregivers about how to reinforce what children have learned, library officials said.

Libraries around the country have expanded story time and other children’s programs in recent years, attracting a new generation of patrons in an age when online offerings sometimes make trips to the book stacks unnecessary. Sari Feldman, president of the American Library Association, said such early-literacy efforts are part of a larger transformation libraries are undergoing to become active learning centers for their communities by offering services like classes in English as a second language, computer skills and career counseling.

Ms. Feldman said the increased demand for story time was a product, in part, of more than a decade of work by the library association and others to encourage libraries to play a larger role in preparing young children for school. In 2004, as part of that effort, the association developed a curriculum, “Every Child Ready to Read,” that she said is now used by thousands of libraries.

The New York Public Library is adding 45 children’s librarians to support story time and other programs, some of which are run in partnership with the city government. It has also designated 20 of its 88 neighborhood branches, including the Fort Washington Library, as “enhanced literacy sites.” As such, they will double their story time sessions, to an average of four a week, and distribute 15,000 “family literacy kits” that include a book and a schedule of story times.

“It is clear that reading and being exposed to books early in life are critical factors in student success,” Anthony W. Marx, president of the New York Public Library, said. “The library is playing an increasingly important role in strengthening early literacy in this city, expanding efforts to bring reading to children and their families through quality, free story times, curated literacy programs, after-school programs and more.”

For its part, the Queens Library plans to expand a “Kick Off to Kindergarten” program that attracted more than 180 families for a series of workshops last year. Library officials said that more than three-quarters of the children who enrolled, many of whom spoke a language other than English at home, developed measurable classroom skills.

At the Fort Washington Library recently, children in strollers were lined up outside when the branch’s door opened at 10 a.m., a half-hour before story time. Ms. Asbury, the library clerk, rushed around to handle the strollers while reminding their occupants to take their time getting out and up the stairs.

Upstairs, Nikki Griffin, a children’s librarian, used a book about Halloweento have the boys and girls count pumpkins, then got them to rise for a foot-stomping rendition of “We Are the Dinosaurs,” a song by the Laurie Berkner Band.

Meredith Ryness, 33, a mother of three, said that though her children loved the dinosaur song, the family did not play it in their apartment because they had downstairs neighbors. “Especially in the winter, this is a lifesaver,” she said. “They can make all the noise they want.”

Afterward, Ms. Ryness added, they check out as many as 20 books for their own story time at home.

Nelly Sanchez, a child-care worker from Washington Heights, in Upper Manhattan, said she had taken her charges to libraries in Battery Park City, Brooklyn and the Bronx just for the story times. Ms. Sanchez, 43, said that now when she was interviewed by parents, they emphasized the importance of reading.

“It used to be, ‘What do you do when the child cries?’” she said. “Nowadays, they’re like, “We would like to have library time.’”

Correction: November 6, 2015

An article on Monday about the increased demand for toddler story time at public libraries across New York City misstated the designation of 20 New York Public Library branches that will hold extra story time sessions and distribute literacy kits. They are enhanced literacy — not literary — sites.