The bodies of the Vietnamese nationals were discovered by emergency services at an industrial estate in Grays, Essex, shortly after the lorry arrived on a ferry from Zeebrugge in Belgium in the early hours of October 23 last year.
Among the men, women and children were 10 teenagers, two of them 15-year-old boys.
Five men charged following an investigation by Essex Police appeared for a virtual hearing at the Old Bailey before Mr Justice Sweeney.
Maurice Robinson, 25, of Craigavon in Northern Ireland, had previously pleaded guilty to conspiracy to assist unlawful immigration and acquiring criminal property.
During the hearing on Wednesday, the truck driver also admitted 39 counts of manslaughter on or before October 24 last year.
He denied a further charge of transferring criminal property.
Robinson appeared at court via video link alongside four other co-defendants.
British Romanian Gheorghe Nica, 43, of Mimosa Close in Langdon Hills, denied 39 counts of manslaughter.
He also denied one count of conspiracy to assist unlawful immigration between May 1 2018 and October 24 2019.
Romanian national Alexandru-Ovidiu Hanga, 27, of Hobart Road in Tilbury, denied a charge of conspiracy to assist unlawful immigration.
Christopher Kennedy, 23, of Corkley Road in Darkley, Co Armagh, Northern Ireland, has previously denied conspiracy to assist unlawful immigration
Valentin Calota, 37, of Cossington Road in Birmingham, was not asked to enter a plea to the charge of conspiring to assist unlawful immigration.
Prosecutor William Emlyn Jones QC said a human trafficking conspiracy charge was being dropped in relation to Kennedy and Robinson.
He asked for three weeks to decide whether to proceed with a trial against Robinson on the outstanding charge he faced.
The other defendants face a trial at the Old Bailey lasting up to eight weeks from October 5.
The hearing was conducted virtually with most lawyers and court reporters attending by Skype.
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