Andrew has had a tumultuous end to his year after his disastrous appearance on the BBC's Newsnight programme, which has left his reputation in tatters.
His attempt to explain his relationship with convicted sex offender Jeffrey Epstein backfired and he was heavily criticised for showing a lack of empathy towards Epstein's victims and remorse over his friendship with the disgraced financier.
In the interview, the duke denied claims that he slept with Virginia Giuffre, one of Epstein's victims, on three separate occasions, twice while she was under age.
Andrew drove himself to the annual event at the palace.
Other guests included the Duchess of Cambridge, who was seen at the wheel of her car, with Princess Charlotte in the back seat.
Andrew has stepped down from official public duties for the foreseeable future, following the backlash from his television appearance in November when he was interviewed by presenter Emily Maitlis.
The duke has only been seen in public a handful of times since the interview and a large number of his major patronages have accepted his resignation and others have distanced themselves from him.
The Duke and Duchess of Sussex are on an extended break, with rumours suggesting they are overseas with Meghan's mother, Doria Ragland, and are not expected to attend the royal gathering.
Kate waved to waiting photographers as she drove through the palace gates, with Prince Louis on the back seat next to sister Charlotte.
The Duke of Cambridge arrived driving a separate vehicle with son Prince George in the front passenger seat.
The Prince of Wales, who earlier attended a service in east London to remember victims of the Sri Lankan Easter terrorist bombings, was chauffeur-driven to the palace.
Other arrivals included the Princess Royal and her husband, Vice Admiral Sir Tim Laurence, Princess Beatrice, and Zara and Mike Tindall.
The Countess of Wessex drove herself to the private event with her daughter Lady Louise Windsor sitting beside her.
Her husband, the Earl of Wessex, arrived separately, driving a vehicle with his son, Viscount Severn, in the back seat.