It says there were "multiple failings" in how both events were handled, despite an official report clearing the Met of heavy-handedness.
New legislation going through Parliament has triggered a debate about the extent of police powers.
Ministers say it will enable officers to better manage demonstrations.
Scotland Yard faced a barrage of criticism in March, including calls for Commissioner Dame Cressida Dick to resign, after a vigil to remember Ms Everard, ended in physical clashes between police and protesters in south London.
Hundreds had gathered to remember the 33-year-old - who was found dead after she went missing while walking home - despite police ruling the event illegal under lockdown restrictions.
And in the same month in Bristol, the Kill the Bill protest against government plans to give police greater powers to control demonstrations, turned into a riot after around 500 people marched on Bridewell police station, setting fire to cars and attacking the building.
Campaigners say police tactics in handling both events are examples of why the Police, Crime, Sentencing and Courts Bill (PCSC) being debated in the Commons next week, needs to be changed.
The All Party Parliamentary Group (APPG) on Democracy and the Constitution has been looking in to both cases.
The group says that both the Metropolitan Police and Avon and Somerset Police wrongly applied "ambiguous" lockdown laws and "failed to conduct a proper assessment of the proportionality of their actions".
The inquiry's chairman, Labour MP Geraint Davies, said: "The police must not become the enforcement agency of the state against those who choose to publicly and collectively call for change - political, economic, social or environmental."
He added: "Parliament must protect our freedoms and reject attempts to increase police power and restrict our right to peaceful protest.
"The police should help to facilitate the expression of peaceful protest and not drive opposition underground."
Among the amendments the MPs are proposing to the new public order laws are scrapping powers to limit the right to peaceful demonstrations and introducing a code for policing demonstrations.
The code would place a "duty" on police to facilitate peaceful protest and allow people to bring legal action against forces if breached, the report added.
But some have questioned what authority APPGs have to conduct inquiries and publish reports on contentious issues.
That is because unlike parliamentary select committees, APPGs have no official status and are informal, cross-party groups of MPs and Lords who share a common interest in a particular subject or policy area.
Conservative MP Kieran Mullan, who sits on the Justice Select Committee, said the APPG's report "should be left to collect dust on a shelf".
"It is clear to me that this whole 'inquiry' is an insult to the word and risks bringing the notion of a Parliamentary inquiry into disrepute.
"Anyone reading it will see it for what it is. A partisan exercise by a bunch of people who have their own agendas."
He added: "The facts that they cannot avoid including are clear.
"The organisers wanted the police to tell them in advance they could hold a protest no matter the circumstances.
The police wouldn't do that. They went to Court to try and force the police to do that and they lost.
"Even then, they were actually allowed to have several hours of their event without police interference."
The government says public order laws are out of date and need to be overhauled.
It says the measures proposed in the Bill would "in no way curtail on the right to peaceful protest".
They would, instead, allow officers to "better manage demonstrations so that legitimate protest groups can make their voices heard without disrupting the lives and livelihoods of others".
The legislation, which was included in the Queen's Speech, will be debated by MPs on Monday.