Video report by ITV News UK Editor Lucy Manning
The Press Complaints Commission (PCC) said it had received 384 complaints by 4pm on Wednesday relating to Daily Mail articles about Ed Miliband's father.
The row between Mr Miliband and the Daily Mail could influence politicians deciding on a new system of press regulation, a national newspaper editor has warned.
Rival proposals are being considered by the Privy Council, which next meets on October 9, and The Independent's editor Chris Blackhurst said it would be hard to "ignore what's been going on".
Mr Miliband insisted that his dispute over the Daily Mail's claim that his father "hated Britain" was not about regulation but "responsibility and right and wrong".
Mr Blackhurst said he hoped the row would not influence the Privy Council but accepted it had "driven a wedge between politicians and the press".
In response to the Leveson Inquiry the leaders of the three main political parties agreed to set up a new press watchdog by royal charter with powers to impose million-pound fines on UK publishers and demand upfront apologies.
But the newspaper industry rejected the idea of "state-sponsored regulation" and put forward a rival plan, which is being examined by a sub-committee of the Privy Council.