Boris Johnson and Rishi Sunak have pledged to stay in post despite being fined by police for breaking lockdown rules in Downing Street in June 2020.
The prime minister, the chancellor and the PM's wife all received fixed penalty notices for attending a birthday gathering for the PM in No 10.
As a result, Mr Johnson became the UK's first serving prime minister to be sanctioned for breaking the law.
All three apologised, but both the PM and Mr Sunak rejected calls to resign.
Mr Johnson said he felt "an even greater sense of obligation to deliver on the priorities of the British people", while the chancellor said he was "focused on delivering for the British people".
But bereaved families said there was "simply no way either the prime minister or chancellor can continue" in their jobs, calling their actions "truly shameless".
Amanda McEgan, whose daughter Isabel died during the pandemic, said the prime minister's conduct during the lockdowns made restrictions on the 19-year-old's funeral "more hurtful".
Opposition parties accused Mr Johnson and Mr Sunak of lying to the public about their attendance at the Downing Street gathering, with Labour leader Sir Keir Starmer and the first ministers of Scotland and Wales demanding they step down.
The SNP and the Liberal Democrats also led calls for Parliament to return from its Easter break so the PM and chancellor could face questions from MPs.
But a raft of cabinet ministers publicly back Mr Johnson and Mr Sunak - including Foreign Secretary Liz Truss, who said they were "delivering for Britain on many fronts".
The fines come as part of an investigation by the Metropolitan Police into illegal parties held in Downing Street and across Whitehall during Covid lockdowns in 2020 and 2021.
The force is looking into 12 parties overall, and has already issued more than 50 fines, with more expected to come.
Speaking on Tuesday, the PM said he accepted "in all sincerity that people had the right to expect better" from him.
He claimed the event he was fined for attending - a gathering in the Cabinet Room to mark his birthday - was "brief" and lasted "less than 10 minutes".
He added: "In all frankness, at that time it did not occur to me that this might have been a breach of the rules.
"But, of course, the police have found otherwise and I fully respect the outcome of their investigation."
A spokesperson for the prime minister's wife, Carrie Johnson, said she had accepted the police's findings and "apologises unreservedly."
Mr Sunak later released a statement, saying: "I understand that for figures in public office, the rules must be applied stringently in order to maintain public confidence.
"I respect the decision that has been made and have paid the fine."
But the "full" and "unreserved" apologies did not silence critics of the PM and chancellor.
Labour's deputy leader, Angela Rayner, tweeted: "You made the rules. You broke your own law. Just go."
The SNP's Westminster leader, Ian Blackford, said the two leading Tories had "insulted the millions of people who faithfully followed the rules".
Liberal Democrat leader Sir Ed Davey said: "This is a government in crisis neglecting a country in crisis."
Plaid Cymru's Commons leader Liz Saville Roberts accused Mr Johnson and Mr Sunak of having an "arrogant sense of exceptionalism and a belief in their own entitlement".
"If they have any honour they will resign," she said.
And Green Party co-leader Adrian Ramsay said that both had broken "the very laws they made to try and keep this country safe in a pandemic and must now resign".
However, Work and Pensions Secretary Therese Coffey said it was "important for the country that [the PM] and the chancellor get on with the job the country elected them to do."
Veteran Conservative MP Sir Roger Gale, who had previously called for Mr Johnson to resign, said the situation was "serious" and that the PM had "effectively misled the House of Commons".
But, he said, now was not the time to "unseat" Mr Johnson amid Russia's invasion of Ukraine.
Writing in the Telegraph, former Conservative Party leader Iain Duncan Smith said the revelation of rule-breaking had been "astonishing and angering", but added: "We are facing two national calamities, the war in Ukraine and the cost-of-living crisis.
"The sheer scale and urgency of these issues place a premium on leadership, which is why… this country's leadership must be allowed to get on with the job."
Reports of parties being held in Downing Street during lockdowns first emerged in December last year, with Mr Johnson initially insisting that "guidelines were followed at all times".
Mr Sunak also told the Commons: "I did not attend any parties."
The PM then ordered an inquiry into the allegations of rule-breaking, led by senior civil servant Sue Gray.
The Met initially said it would not retrospectively investigate the allegations unless "significant evidence" of a regulation breach came forward.
But after Ms Gray passed information to officers, they launched their own inquiry.
Her full report will not be released until the Met have concluded their investigation.