The city’s toniest pre-Ks bar parents from giving holiday gifts to teachers and directors — but moms and dads are so desperate to curry favor with the staffers who recommend their kids to future schools that they’ve turned to smuggling over-the-top “tips” to them.

Tiffany boxes, Birkin bags, Hermès scarves, diamond bracelets and even cash are standard offerings for the employees, parents and workers told The Post.

“It’s just one more way of protecting your child,” said one parent whose children attended pre-school at the Central Synagogue on Lexington Avenue off East 55th Street. “It’s just one more way of competing in NYC.”

Dana Haddad, a former private-school teacher, said parents would sneak gifts to her in creative ways.

“Parents met me outside of school to give [them] to me,” she recalled. “The rule was that it had to be a homemade gift, so once, I got a cookie jar that was actually homemade — but it was filled with $500 in cash.”

School officials began banning the gifts a few years ago because they were becoming extravagant.

When asked about parents continuing to give gifts, Ellen Davis, director of the nursery school at Temple Emanu-El on the Upper East Side, flat out denied it.

“That’s absolutely not true. They’re not allowed,” she insisted.

But parents said Emanu-El is among the worst offenders in the gifting competition. One year, a parent gave both a head teacher and her assistant gifts from Hermès — and the main instructor threw a hissy fit because the items were so close in value, a parent said. The gifts were eventually exchanged by the parent to reflect the workers’ different levels of experience.

A former admissions director at Madison Avenue Presbyterian Day School said she was showered every holiday season with spa gift cards, bottles of wine and blue Tiffany boxes.

“I had a lot of really nice dinners at fancy restaurants with $1,000 bottles of wine,” she reminisced. “The most powerful people in the city kissed my ass. It was lovely.”

Parents said they fear if they don’t pony up, their kids won’t get good enough recommendations to get into the best schools.

The premiere nursery school in Manhattan, the 92nd Street Y, actually embraces the gift culture; one parent recalled receiving a list of things that the teachers didn’t want for the holidays.

“I remember being surprised that the thing they would not like to get was a framed picture of your child,” the parent huffed. “It felt cold and even a little mercenary.”

Social researcher Wednesday Martin, who is writing a book about the city’s elite called “The Primates of Park Avenue,” said, “[Parents] are sometimes confused that these teachers are not service providers.”