The gatherings, first reported by The Telegraph, took place on 16 April 2021 and went on until the early hours.
The PM'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 Mr 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 North, Pensions Minister Guy Opperman said Mr 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.
Mr Opperman said the prime minister should continue in post while the investigation into government gatherings is completed. But he said Mr 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 Ms Gry.
After No 10 apologised for the events, Labour leader Sir Keir Starmer said: "This shows just how seriously Boris Johnson has degraded the office of prime minister.
"The Conservatives have let Britain down. An apology isn't the only thing the prime minister should be offering the Palace today.
"Boris Johnson should do the decent thing and resign."
At this point all Downing Street can do is apologise.
The plan was to buy time waiting for Sue Gray's report but even that became untenable as Boris Johnson was forced to say sorry to the Commons and now to the Queen.
The problem for the prime minister and his team is that they are not in control of this situation, they don't know what's coming next and can only react.
The absurd detail about filling a suitcase with wine in the latest leak has prompted a raft of memes mocking Mr Johnson and his staff. No politician wants to be the object of ridicule.
For now it's clear they're trying to insulate the prime minister, pointing out that he was away when these latest antics took place.
There's no sign his cabinet is plotting against him, but the danger hasn't passed.
Government officials apologised to the Palace in a phone call.
Asked why No 10 had apologised rather than Boris Johnson himself, his spokesman told reporters: "Well, again, the prime minister said earlier misjudgements have been made and it's right people apologise, as the PM did earlier this week."
The spokesman says he could not prejudge Ms Gray's inquiry, but added: "We acknowledge the significant public anger, it was regrettable this took place a time of national mourning."
Liberal Democrat leader Sir Ed Davey called on Mr Johnson to apologise personally to the Queen "for the offence he's caused her and millions around the country mourning for loved ones".
The two parties last April involved around 30 people in total, and are reported to have converged at some point in the Downing Street garden, where they continued past midnight.
Staff were reportedly sent to a nearby shop with a suitcase, that was brought back "filled with bottles of wine".
They were both leaving parties: one for the PM's then director of communications, James Slack, and the other for one of the PM's personal photographers.
Mr Slack, who is now deputy editor of The Sun newspaper, has apologised for the "anger and hurt" caused by the party, and acknowledged it "should not have happened at the time that it did".
At the time, England was under "step two" restrictions that stipulated people could not socialise indoors, except with those from their household or support bubble.
People could socialise outdoors in groups of up to six people or two households.
The government is facing mounting pressure over several events that are alleged to have been held during lockdowns. Here is what we know about them and the restrictions in place at the time:
Boris Johnson announced a plan to take the “first careful steps" out of the lockdown that began in March 2020. But he said people should continue to "obey the rules on social distancing and to enforce those rules we will increase the fines for the small minority who break them”.
Legal restrictions at the time said you could not leave your house without a reasonable excuse and government guidance was that you could meet one person outside of your household in an outdoor setting while exercising.
A photo from May 2020 showed the prime minister and his staff with bottles of wine and a cheeseboard in the Downing Street garden. When asked about it, Boris Johnson said, “those people were at work talking about work”.
About 100 people were invited by email to “socially distanced drinks in the No 10 garden” on behalf of the prime minister’s principal private secretary, Martin Reynolds.
Witnesses told the BBC the PM and his wife were among about 30 people who attended.
Boris Johnson has confirmed he attended the event, saying he was there for 25 minutes and “believed implicitly that this was a work event”.
Boris Johnson announced plans for a “significant return to normality" in England by Christmas "through targeted, local action” instead of national lockdowns.
But he added that the timetable relied on “every one of us staying alert and acting responsibly”.
With cases of coronavirus rising again, the prime minister told people in England that “we are once again asking you to stay at home” as a new national lockdown began.
He said people should only leave their homes “for work if you can’t work from home, for education, and for essential activities and emergencies”. Indoor gatherings with other households were banned, unless they were for work purposes.
Sources told the BBC that Downing Street staff members attended a gathering with Carrie Johnson in the flat where she and the prime minister live. A spokesman for Mrs Johnson denies the party took place.
A leaving event was held for No 10 aide, Cleo Watson, where people were drinking, and Mr Johnson made a speech, according to sources.
The second national lockdown ended after four weeks but Boris Johnson replaced those restrictions with “tough tiers to keep this virus down”.
London was placed in tier two, which banned two or more people from different households from meeting indoors, unless “reasonably necessary” for work purposes.
The Department for Education has confirmed it had an office gathering to thank staff for their work during the pandemic. It says drinks and snacks were brought by those who attended and no outside guests or support staff were invited.
The Conservative Party has admitted that an “unauthorised gathering” took place at its HQ in Westminster. It was held by the team of the party's London-mayoral candidate, Shaun Bailey, who has since stepped down as chair of the London Assembly police and crime committee. The Metropolitan Police is to speak to two people who attended the party.
Multiple sources have told the BBC there was a Christmas quiz for No 10 staff last year. A photo - published by the Sunday Mirror - showed Boris Johnson taking part and sitting between two colleagues in No 10. Mr Johnson has denied any wrongdoing.
London moved into the highest tier of restrictions and Matt Hancock, who was health secretary at the time, said it was important “everyone is cautious” ahead of the festive period.
The Department for Transport apologised after confirming reports of a party in its offices that day, calling it “inappropriate" and an "error of judgment” by staff.
A leaving party was held at the Cabinet Office for the outgoing head of the civil service Covid taskforce - the team responsible for drawing up coronavirus restrictions.
Kate Josephs, now chief executive of Sheffield City Council, apologised for the event, saying she was “truly sorry that I did this and for the anger that people will feel as a result”.
Downing Street originally denied a report by the Daily Mirror that a party took place in Downing Street.
However, a video obtained by ITV News showed the prime minister's then-press secretary Allegra Stratton, joking about reports of an event, saying: “This fictional party was a business meeting and it was not socially distanced.”
Lockdown restrictions were eased in England, with pubs and restaurants allowed to reopen with outdoor service only.
However, working from home continued to be recommended and socialising indoors with people from other households was not allowed. Meeting others outdoors was limited to groups of six people or two households.
Two parties were held by Downing Street staff at No 10, the night before Prince Philip's funeral.
One of the events was a leaving party for the PM's then director of communications James Slack, who has apologised for the event and acknowledged it “should not have happened at the time that it did”.
Boris Johnson was not at either party.