The UK-based political consulting firm is accused of using the data of 50 million Facebook members to influence the 2016 US presidential election.
Damian Collins, the chairman of the Commons inquiry into fake news, accused Facebook of “misleading” the committee.
London-based firm Cambridge Analytica denies any wrongdoing.
Mr Collins said earlier in the week that he wanted to hear from Mr Zuckerberg but has now put the request into writing.
In the letter, he says: “It is now time to hear from a senior Facebook executive with the sufficient authority to give an accurate account of this catastrophic failure of process.”
It comes after the UK’s Information Commissioner Elizabeth Denham said she would be applying to court for a warrant to search the offices of Cambridge Analytica.
In the letter, Mr Collins criticised other Facebook executives who had already given evidence to his committee.
“The committee has repeatedly asked Facebook about how companies acquire and hold on to user data from their site, and in particular about whether data had been taken without their consent,” he wrote.
“Your officials’ answers have consistently understated this risk, and have been misleading to the committee.”
Now he wants to hear from a senior Facebook executive “with the sufficient authority”, adding: “Given your commitment at the start of the New Year to “fixing” Facebook, I hope that this representative will be you.”
Mr Collins has asked for a response to his request by 26 March.
Media captionAlexander Nix, CEO of Cambridge Analytica was filmed by undercover reporters for Channel 4 News
UK Information Commissioner Elizabeth Denham is investigating Cambridge Analytica over claims it used personal data to influence the US election.
Christopher Wylie, who worked with the company, claimed it amassed the data of millions of people through a personality quiz on Facebook called This is Your Digital Life that was created by an academic.
Facebook said Aleksandr Kogan, who created the personality app from which the data had been harvested, has agreed to be audited, but Mr Wylie – who made the claims about the way the data was gathered and used – has declined.
‘Honey traps’
Channel 4 News has also broadcast hidden camera footage in which Cambridge Analytica chief executive Alexander Nix appears to suggest tactics his company could use to discredit politicians online.
Media captionFormer Cambridge Analytica employee Christopher Wylie says the firm planted fake news
In the footage, asked what “deep digging” could be done, Mr Nix told an undercover reporter: “Oh, we do a lot more than that.”
He suggested one way to target an individual was to “offer them a deal that’s too good to be true and make sure that’s video recorded”.
He also said he could “send some girls around to the candidate’s house…” adding that Ukrainian girls “are very beautiful, I find that works very well”.
Mr Nix continued: “I’m just giving you examples of what can be done and what has been done.”
Channel 4 News said its reporter had posed as a fixer for a wealthy client hoping to get a political candidate elected in Sri Lanka.
However, Cambridge Analytica said the report had “grossly misrepresented” the conversations caught on camera.
“In playing along with this line of conversation, and partly to spare our ‘client’ from embarrassment, we entertained a series of ludicrous hypothetical scenarios,” the company said in a statement.
“Cambridge Analytica does not condone or engage in entrapment, bribes or so-called ‘honey traps’,” it said.
Mr Nix told the BBC’s Newsnight programme that he regarded the report as a “misrepresentation of the facts” and said he felt the firm had been “deliberately entrapped”.
On its website, Cambridge Analytica says it can provide data and insights to “drive your voters to the polls and win your campaign” and says it has played a “pivotal role” in winning US presidential races.
Media captionMr Nix spoke to BBC Newsnight before the Channel 4 report was aired on Monday night. He declined to be interviewed after the undercover footage was broadcast
How to protect your data on Facebook
There are a few things to be aware of if you want to restrict who has access to your data.
Keep an eye on apps, especially those which require you to log in using your Facebook account – they often have a very wide range of permissions and many are specifically designed to pick up your data
Use an ad blocker to limit advertising
Look at your Facebook security settings and make sure you are aware of what is enabled. Check the individual app settings to see whether you have given them permission to view your friends as well as yourself.
You can download a copy of the data Facebook holds on you, although it is not comprehensive. There is a download button at the bottom of the General Account Settings tab. However bear in mind that your data may be less secure sitting on your laptop than it is on Facebook’s servers, if your device is hacked.
You can of course, simply leave Facebook, but the campaign group Privacy International warns that privacy concerns extend beyond the social network.
“The current focus is on protecting your data being exploited by third parties, but your data is being exploited all the time,” said a spokeswoman.
“Many apps on your phone will have permission to access location data, your entire phone book and so on. It is just the tip of the iceberg.”
