The anger directed against MPs over Brexit is "not surprising", the PM's adviser, Dominic Cummings, has said.
The former Vote Leave campaign director said the only way the issue of abuse would be solved is if MPs "respect" the result of the EU referendum.
Mr Cummings's remarks came after Boris Johnson defended language he used in Parliament amid criticism from MPs.
The parliamentary tensions have led 120 archbishops and bishops to warn against "further entrenching our divisions".
The intervention followed an ill-tempered debate on Wednesday, as MPs returned to Parliament after the Supreme Court ruled the suspension of Parliament was unlawful.
The prime minister was criticised by a number of MPs for - among other remarks - describing one Labour MP's safety concerns as "humbug" and repeatedly referring to legislation aimed at blocking no-deal as "the surrender bill".
On Thursday, the Commons heard of threats faced by politicians, with independent MP Caroline Nokes describing how someone had called her a "traitor who deserved to be shot" on a walkabout in her constituency.
Speaking at a book launch that evening, Mr Cummings said MPs had spent three years "swerving all over the shop" following the referendum and it was "not surprising some people are angry about it".
He said both Leave and Remain campaigners had received "serious threats" of violence, which he said should be taken seriously.
But he added: "If you are a bunch of politicians and say that we swear we are going to respect the result of a democratic vote, and then after you lose you say, 'We don't want to respect that vote', what do you expect to happen?"
"In the end, the situation can only be resolved by Parliament honouring its promise to respect the result," he said, echoing sentiments expressed by the prime minister in the Commons on Wednesday.
But former Justice Secretary David Gauke told BBC Radio 4's Today programme: "Some of the language from the prime minister this week has clearly made it harder to win support from Labour MPs for any kind of a deal."
Mr Cummings denied that Downing Street was under pressure following the Supreme Court ruling, a series of parliamentary defeats and the backlash against Mr Johnson's comments.
"This is a walk in the park compared to the referendum. We are enjoying this. We are going to leave and we are going to win," he said.
But, when questioned as he left his home in London on Friday morning, Mr Cummings said: "Who said it would be a walk in the park?"
Told that he had made the remark, he replied: "No."
MPs have expressed concern that Downing Street could seek to bypass legislation - passed earlier this month - to block a no-deal Brexit.
The Benn Act - which Mr Johnson has been referring to as the "surrender act" - says the prime minister will have to ask the EU for an extension to the 31 October Brexit deadline if he is unable to pass a deal in Parliament, or get MPs to approve a no-deal Brexit, by 19 October.
What is the Benn Act?
When Mr Johnson talks about the "surrender bill", he is referring to the European Union (Withdrawal) (No. 2) Act, also known as the Benn Act after Labour MP Hilary Benn, who introduced the legislation to the Commons.
The act - which became law earlier this month - stipulates the prime minister will have until 19 October to either pass a deal in Parliament or get MPs to approve a no-deal Brexit.
Once this deadline has passed, he will have to request an extension to the UK's departure date to 31 January 2020 from the EU.
If the EU responds by proposing a different date, the PM will have two days to accept that proposal. But during this two-day period, MPs - not the government - will have the opportunity to reject the EU's date.
Former Prime Minister Sir John Major - who on Thursday accused Mr Johnson of "wilfully" destroying the prospects of a cross-party agreement on Brexit - expressed concern that the government might sidestep the law by suspending the Benn Act until after 31 October.
Sir John said he thought ministers might be planning to do this by passing an Order of Council, which can be approved by Privy Councillors - government ministers - and has the force of law.
Asked if he was looking at using this method to get around the Benn Act, the prime minister said "no".
And senior cabinet minister told the BBC that such a plan would be "too clever by half".
Downing Street has consistently said the government will obey the law, but Mr Johnson has also insisted he will not seek a delay to Brexit, which the act mandates.
Questioned on the government's position, International Development Secretary Alok Sharma told Today: "I'm not going to set out discussions that have occurred in the privacy of cabinet."
He added that the government would "absolutely" comply with the law.
In interviews with the BBC, Mr Johnson acknowledged that "tempers need to come down" in Parliament.
But he added: "I do think in the House of Commons it is important I should be able to talk about the surrender bill, the surrender act, in the way that I did."
Meanwhile, the College of Bishops has called on politicians to "speak to others with respect", adding that the result of the EU referendum "should be honoured".