Ashley Carlson called out homework instructions across a large lawn to her first grader, who was leaving with a babysitter after having practiced cartwheels on the grass, and then turned her attention to her 3- and 4-year-old sons. The two boys, with matching white-blond hair and blue polo shirts, were clambering up a low stone wall and leaning on a painstakingly restored early-1900s ornate iron fence.

Between the familiar parental refrains of “Don’t pull on that!” and “Oh, don’t do that,” Ms. Carlson touted the virtues of the lawn. “We live a block away, and we come here all the time,” she said early this fall. “This is sort of like a backyard. A stylish backyard.”

The 7,600-square-foot garden of the Cooper Hewitt, Smithsonian Design Museum, on East 90th Street just off Fifth Avenue, reopened late last year, the final chapter in a top-to-bottom renovation of the museum.

A lawn that began as the private Manhattan garden of Andrew Carnegie and was later made available for neighbors who paid for a key has since become a hangout for local parents of young children and the people charged with taking care of them. There is no fee to enter, and it is open to the public, no museum visit required. A cafe sells lattes, sandwiches, gelato (popular with the child set) and wine (popular with the chasing-the-child set).

Ms. Carlson said that security guards have had to reprimand her boys just once, reminding them that “they’re not allowed to stand on the artwork.”

On a recent day, the lawn was the bustling play space of nannies pushing strollers, a towheaded preteenager balancing on an orange spinning chair and a preschooler in a pinafore having lunch from a gourmet grocery bag.

“It still manages to be really tranquil even though there are quite a good number of kids running around,” said Marguerite Rogers, who has made the garden a regular stop with her 4-year-old son, Jamie. An elevated pathway, with a brick wall overlooking the main yard and steps to a back entrance (now closed) to the mansion that houses the museum, is lined with colorful yarrow and coral bells, and is one of her son’s favorite places to play, Ms. Rogers said.

Parents said they are drawn not just to the flower-filled pathways, but also to the eclectic mix of Upper East Siders, international tourists and artist types who don’t frequently visit an uptown neighborhood that to some can have an almost suburban feel. “It’s not like when you go to a Gymboree-type place where it’s all mothers and babies,” Ms. Rogers said. Recently, she said, she met and had an actual adult conversation about culture with a Juilliard-trained singer.

Toddlers and high design may seem like strange bedfellows, but there is nothing for children to break or destroy in the garden. “My aim was really to be the friendliest place on Museum Mile,” said Caroline Baumann, the museum’s director, who hopes it will draw more people inside. “The garden, for me, is the gateway drug for design.”

For the parentarazzi who like to artfully document their children’s every move (and even their naps), there is plenty of Insta-bait. The outdoor furniture includes spare tables and chairs (designed by Stefano Giovannoni and Jasper Morrison, and donated by Herman Miller), curvy black-and-white modular “Copabananas” benches in the grass, and Yves Behar ipe wood and aluminum benches on the walkways.

The most popular photo backdrops seem to be the off-balance rotating orange and gray chairs, designed by Thomas Heatherwick. One woman deemed them “much more comfortable than they look” as she swayed front to back and kept an eye on a girl drawing in a notebook in the grass.

Near the Fifth Avenue gate is a blue Ping-Pong table, a selection by Ms. Baumann, who played daily as a child and regularly challenges colleagues to matches. (Maintenance work has blocked parts of the lawn and displaced some furniture this month, but the museum expects to restore it within days.)

As Ms. Baumann, wearing sunglasses with bamboo frames and a short-sleeve knit dress that matched the bright orange of the chairs, surveyed the garden, Lance Somerfeld and a couple of members of his NYC Dads group were wrapping up a long play date. His 1-year-old daughter had spent much of her time pushing around a toy stroller.

“It’s just a beautifully landscaped area,” said Mr. Somerfeld, who added that he has come with his wife as well, to enjoy adult time and a glass of wine. His group meets there regularly to “chain a few blankets and just kind of relax.” With access to bathrooms with changing tables, they sometimes stay for hours.

Ms. Baumann said she is trying to maintain the garden’s quiet, sophisticated vibe. “The museum is monitoring the crowds so that it always remains comfortable,” Ms. Baumann said diplomatically, and some rules have been implemented. “We can’t have soccer games in here. We can’t order in 14 boxes of pizza for your kid’s third birthday.”

The museum has built in some landscape features to try to impede small feet from stamping through growing plants. For example, a foot-high rope was installed around the backyard’s “rockery” — a legacy of the original 1901 design of the space. But it will come down soon, both because the thyme has grown in and the barrier did little to dissuade children, including Ms. Baumann’s own son, Hugo, 19 months old.