Up to 90 email accounts were compromised during the cyber-attack on Parliament on Friday.
Fewer than 1% of the 9,000 users of the IT system were impacted by the hacking, said a parliamentary spokesman.
The hack prompted officials to disable remote access to the emails of MPs, peers and their staff as a safeguard.
The spokesman said the attack was a result of "weak passwords" and an investigation is under way to determine whether any data has been lost.
Both Houses of Parliament will meet as planned on Monday and plans are being put in place to allow it to resume its wider IT services, said officials.
A number of MPs confirmed to the BBC they were unable to access their parliamentary email accounts outside of the Westminster estate following the hacking.
'Passwords for sale'
The spokesman said the parliamentary network was compromised due to "weak passwords" which did not conform to guidance from the Parliamentary Digital Service.
They added: "As they are identified, the individuals whose accounts have been compromised have been contacted and investigations to determine whether any data has been lost are under way."
The incident comes just over a month after 48 of England's NHS trusts were hit by a cyber-attack.
International Trade Secretary Liam Fox said: "We have seen reports in the last few days of even cabinet ministers' passwords being for sale online.
"We know that our public services are attacked so it is not at all surprising that there should be an attempt to hack into parliamentary emails.
"And it's a warning to everybody, whether they are in Parliament or elsewhere, that they need to do everything possible to maintain their own cyber-security."
The latest attack was publicly revealed by Liberal Democrat peer Lord Rennard on Twitter as he asked his followers to send any "urgent messages" to him by text.
The National Cyber Security Centre and National Crime Agency are investigating the incident.