Three months before Olga Lopatina and Emiliano Beltrami make their annual August pilgrimage to Mykonos, they start making reservations.

They make sure to snag tables at Scorpios, a bohemian beach club set on the Aegean Sea, and Lío, the Greek island’s star-studded, cabaret-style restaurant. “We make reservations for six people even before we know which of our friends can come,” said Mr. Beltrami, 45, who lives in Milan with his family and is a real estate entrepreneur. “It’s what you have to do to get a table.”

The most important booking they make (and one of the hardest to get) is for nannies who specialize in late nights to care for their children, ages 2 and 4.

In party hot spots around Europe including Mykonos, where beautiful people guzzle magnums of rosé at oceanside clubs, and Ibiza, where drug-enhanced D.J. sets last until sunrise, agencies have popped up that provide child care at all hours of the day and night.

Many of these babysitters — some clients call them disco nannies — have graduate degrees in education, are fluent in multiple languages, and offer art classes or swim lessons. They arrive at the tourists’ villas or hotels while the parents are getting ready to go out. They will spend the night and sometimes stay until the parents’ hangovers subside the next day. (Parents are expected to furnish a legitimate place for nannies to sleep, not just a chair in the corner of a hotel room.)

For parents who can afford them, these services provide a rare opportunity to let loose. Assured their children are in good hands, many even get to experience a glimmer of their former lives, before parental responsibilities took hold. For their children, it’s an opportunity to interact with locals and get to do activities their parents may not be able to provide.

Before having children, Ms. Lopatina, 38, a former model, and her husband used to drink and dance under the stars, by the sea, all night long.

“We used to live it up in Mykonos, the way everybody wants to,” Mr. Beltrami said. “We partied as much as possible, as much as our bodies allowed us to do.”

Two kids later, with the help of the nannies, they still manage to go out alone at least half their nights in Mykonos. “Some nights we will meet up with friends or try the new trendy restaurants or clubs,” he said. “When we are out alone, we try to feel a little younger.”

These nanny agencies have enjoyed a boom in business as travel restrictions lift and parents who were home with their children nonstop during the early pandemic are eager for a summer break.

“Business is up 15 percent,” said Fani Kotoula, who owns Ms. Lopatina and Mr. Beltrami’s preferred agency, Mykonos Best Nannies. “This summer is extremely busy, and many want overnight nannies so they can go to dinner and clubbing.”

She said demand was so high that some people book up to a year in advance. “I had one client who just booked a nanny for the baby that is still in her belly,” Ms. Kotoula said.

Paula Felicó Marí, an owner — along with María Lopez Muñoz — of Kids & Co., a nanny agency in Ibiza, estimated that business was up 20 to 30 percent from last year. “Many of our nannies stay with children from 10 p.m. in the night to 6 a.m. in the morning, or they will stay until 2 p.m. so the parents can sleep and have some rest,” she said. She said most of her clients are from Europe but she gets requests from Americans, Australians and other nationalities as well.

New hotels are also joining the trend, offering 24-hour babysitting services. Six Senses Ibiza, which opened in July 2021, has a dedicated play area where nannies can bring children while their parents sleep in. Kalesma Mykonos, where every room has a private pool, also opened last year and tells clients it can secure sitters for them any hour, day or night.

Nanny services cost around 20 to 30 euros an hour (about the same in dollars).

Of course, having clients who are partying hard, especially those drinking or using other substances, carries some risk.

Christina Bosmans, the owner of Kids in Ibiza, a babysitting agency on the island, said her best clients are self-aware enough to know how long they will be out and whether they need an overnight nanny. “If you say you are just going out to dinner, but then you come home at 7 a.m., that causes a problem with our scheduling,” she said. “This year in particular seems to be more complicated. People seem to be a bit more flaky.”

“We could write a book about the things that we have seen, the situations that our nannies have found themselves in and the questionable parenting choices adults have made whilst holidaying in Ibiza,” she added in an email. She has started screening clients more rigorously and is only taking on new ones who seem respectful.

Ms. Kotoula of Mykonos Best Nannies said she could recall only one negative experience with intoxicated parents. “Only one time there was this couple, and they were so tired they went directly to the bed and couldn’t pay the babysitter,” she said, laughing. “So the nanny had to wait there until they woke up.”

Alba Sellés, 26, who is originally from Elche, a city in southeast Spain, but who lives in Ibiza during the summer to work as a nanny, said most parents were respectful and grateful. “The parents are always so appreciative,” she said. “We are helping them have a little break and enjoy themselves.”

She regularly stays with her charges overnight and then until 4 p.m. the next day. “I’ll wake up with the kids, have breakfast, play games and sometimes go swimming with them,” she said. “I stay with them until the parents are feeling better and ready to be with the kids.”

Ms. Sellés said she also meets parents at restaurants while they enjoy a morning-after recovery lunch. She will watch the children at a separate table as their parents nurse bloody marys and eat breakfast sandwiches. Some restaurants on the island including El Chiringuito, a beachfront restaurant with live D.J.s, a cigarette vending machine and a jamón-carving station, have dedicated areas for children that are supervised. There are beach toys, playground equipment and other children to keep the little ones entertained for hours.

Some parents say the nanny services they get on vacation are better than what they can find at home.

“We don’t have this setup in England, so we hardly go out at home,” said Lauren Jones, 37, who lives in London and works for a construction company owned by her husband, Nick Jones, 38. “All of our date nights are in Ibiza.”

The couple has a 3½-year-old and a newborn. So when they come to Ibiza — they try to spend up to six months there a year — they feel grateful they have a babysitting service they can trust.

“We are very lucky that when we go to Ibiza we get a little adult time,” Ms. Jones said. In fact, they try to go out two to three times a week while in Spain — not clubbing, per se, but to dinners that start at sunset and can last until 2 or 3 in the morning.

Mr. Jones said he felt as if he could have a few more drinks than usual knowing his children were in safe hands. “It’s easy to get carried away in Ibiza, so it’s nice to know we can do this, and our children are taken care of,” he said. His favorite evening is to start with gin and tonics and move on to wine or cava sangria, a specialty on the island.

The couple looks forward to when their children are a little older and they can use their agency’s recovery service, where the nannies stay until the next morning to let parents sleep in after a night out.

“We haven’t used it yet, but never say never,” Ms. Jones said. “It’s nice that we can go out, and do whatever we want, and have peace of mind.”