Speaking amid the controversy over the Prime Minister’s dismissal of calls for him to temper his “inflammatory” language, Mr Cummings said the only way to address threats made to MPs was to “respect the result” of the referendum.
Mr Cummings also accused many MPs of being "disconnected" from what people think in the "real world".
His comments come as the Church of England criticised MPs' language as "not worthy of our country".
In a statement the College of Bishops urged people to "speak to others with respect", adding that the result of the EU referendum "should be honoured".
The row comes after fiery exchanges in the Commons, in which Boris Johnson was urged to temper his language - including the description of attempts to block a no-deal Brexit as the ‘surrender act’ - amid concerns it is provoking threats to MPs.
The Prime Minister dismissed a Labour MP's complaint that his language risked provoking attacks as "humbug" and sparked anger when he said the best way to honour murdered MP Jo Cox was to “get Brexit done”.
Asked after the book launch of a Vote Leave backer on Thursday night whether he blamed MPs for the abuse, Mr Cummings criticised them for not respecting the result of the 2016 referendum, adding: "It is not surprising some people are angry about it."
He said both Leave and Remain campaigners have had "serious threats" of violence and both should be taken seriously.
He added: “In the end the situation can only be resolved by Parliament honouring its promise to respect the result.”
Mr Cummings also described the current political chaos as "a walk in the park" compared to the referendum during which he was Vote Leave campaign director, adding: “We are enjoying this, we are going to leave and we are going to win.”
Parliamentary tensions have prompted the College of Bishops to issue a statement warning against “further entrenching our divisions”.
It said: “We call on politicians to adhere rigorously to the rule of law and on all to respect and uphold the impartiality of the courts and our judiciary.
“It is easy to descend into division and abuse - climbing out and finding unity again takes far longer.
“Further entrenching our divisions, whether from uncertainty or from partisanship, is not worthy of our country nor the leadership we now need.”
Critics of Mr Johnson’s language include Labour leader Jeremy Corbyn, who said the PM's use of terms like “betrayal” and “surrender” risked driving some people to “unbelievable extremes”.
He said: “The Prime Minister's language is encouraging people to behave in a disgraceful and abusive way to other public figures.”
Mr Johnson’s pro-Remain sister also hit out at his “strongman” tactics, accusing him of using the Commons despatch box as a “bully pulpit”.
She said: “What we are seeing is an executive that is so keen to deliver Brexit in any shape or form... they will do anything to justify that end.”