"We have agreed to that action and are encouraged by the timeline for resolution of these matters, which should be concluded by early November," Bridges said.
"National will remove the videos by 5pm today."
Bridges faced further restrictions if he had continued to defy Mallard's ruling, with 5pm today being the first deadline.
Mallard welcomed Bridges' announcement.
He had told the House that he had wanted a rapid resolution of the issue three weeks ago, when the complaint was first laid, and that defiance of his ruling was an impediment to that.
"I am pleased Mr Bridges has agreed with me," Mallard said.
The issue came to a head following a Labour complaint to Mallard three weeks ago over a National Party video that showed Labour MP Deborah Russell's speech in the House about wellbeing that included remarks about Greek philosophy.
Text accompanying the video read: "Still not sure what Labour's Wellbeing Budget means?"
Mallard ruled that the video violated Standing Orders about the use of Parliament's official television coverage of the House.
Standing Orders require the permission of all MPs shown for footage used for political advertising.
Reports that use extracts of official coverage must also be "fair and accurate".
Mallard ordered all political parties to take down any videos made this year featuring MPs in the House who had not given their consent - which only really affects National.
But National refused, and Bridges called Mallard's ruling "a chilling move designed to stop freedom of expression".
On Tuesday Mallard punished Bridges for defying his ruling around the use of Parliament TV footage, but declined using any serious sanctions.
He decided to restrict the number of National's supplementary questions that Bridges could use during Question Time, adding that these would be further restricted with every week that Bridges continued to defy the ruling.
Mallard could have referred Bridges to the privileges committee, which could have recommended a number of punishments, or decided to name him, which would have banned him from participating in parliamentary proceedings and docked his pay.
Mallard said though the breach was fairly trivial, it was his duty to uphold the authority of the House.
"I ruled that videos using Parliament TV footage had to come down by 5pm on Friday, 27 September. The Leader of the Opposition has defied that ruling.
"I cannot choose to selectively enforce the Standing Orders agreed by the House, any more than the Leader of the Opposition can choose to ignore Standing Orders he does not agree with."
The Standing Orders committee, which Mallard chairs, is reviewing the current rules.
It will be up to the committee to change the rules, and if so, when those changes might apply.
