It was a fifth medal in five Olympics for 30-year-old Daley, who won gold with Matty Lee in the same event at the pandemic-postponed Tokyo Games three years ago.
Lian and Yang, who have won at the last three world championships, established their dominance with their first dive and never looked like giving up the lead.
They finished with a score of 490.35, ahead of Daley and Williams on 463.44, and Canada's Rylan Wiens and Nathan Zsombor-Murray on 422.13.
Daley and Williams were locked in a battle for second with the Canadian pair but pulled away over the second half of the competition.
Daley made his Olympic debut as a 14-year-old at the 2008 Beijing Games.
He paid tribute to his late father Rob, who died of cancer in 2011, after winning gold in Tokyo.
He was one of Britain's flag-bearers at the Paris Games along with rower Helen Glover.
Daley's long-time coach Jane Figueiredo told the BBC: "Excited, fearless, just over the moon for both of them.
"There's been a lot going on behind the scenes for both of them, to come and achieve a silver medal is absolutely brilliant. China was just too good.
"We hung in there, but it was brilliant."
