The gatherings, first reported by The Telegraph, took place on 16 April 2021 and went on until the early hours.
The prime minister's spokesman said it was "deeply regrettable that this took place at a time of national mourning".
Boris Johnson was not at either party - but he faces questions over alleged Covid rule-breaking at No 10.
The latest party revelations led to a backlash from opposition parties, who contrasted the behaviour of No 10 staff with pictures of the Queen sitting alone at the Duke of Edinburgh's funeral, due to Covid restrictions.
Labour, the Liberal Democrats and the SNP are all calling for Johnson to resign after he admitted attending a drinks party in the Downing Street garden during lockdown on 20 May 2020.
And Andrew Bridgen has become the fifth Conservative MP to publicly declare they have written to the chairman of the 1922 Committee - which organises Tory leadership contests - to say they have no confidence in the prime minister.
Fifty-four Conservative MPs have to write a letter to trigger a vote.
Speaking to BBC's Politics Look North, Pensions Minister Guy Opperman said Johnson's behaviour had been "unacceptable".
The Conservative MP said he felt particularly emotional about the party held in the Downing Street garden on 20 May because around that time Covid restrictions had prevented him from being with his wife and twins when they were ill in hospital. The two boys subsequently died shortly after birth.
Opperman said the prime minister should continue in post while the investigation into government gatherings is completed. But he said Johnson needed to "change his ways" and run Downing Street "in a very, very different way".
Meanwhile, the former head of the government's Covid taskforce has apologised for having leaving drinks in the Cabinet Office when she left the civil service on 17 December 2020.
Kate Josephs, who is now the chief executive of Sheffield City Council, said she was "truly sorry" for "the anger that people will feel".
In a statement on Friday, she added that the drinks event would also now form part of the investigation, which is being conducted by senior civil servant Sue Gray.
Sources have confirmed that this was a separate event to a quiz held for the private office of Simon Case, the UK's top civil servant, on the same evening.
The two staff get-togethers for Downing Street staff on 16 April last year happened at a time when Covid restrictions banned indoor socialising.
They have also been added to the list of gatherings being reported on by Gray.