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David Smith, whose 14-year-old daughter Hannah was found dead in her bedroom earlier this month, described Ask.fm's new measures as "a good thing".
But he said that in making changes, the site had "admitted that their website was dangerous for teenagers."
"It's now time for the Government to step up and start regulating these websites so that a 45-year-old lorry driver doesn't have to do it, because I haven't got time to do it any more," he added.
"It's not just Ask.fm, it's the whole internet. The Government actually do need step up and start regulating this internet to make the internet a safer place."
NSPCC safer technology expert Claire Lilley said: "However, these changes alone are not going to solve the problem of online bullying. And, while they are being implemented, children and young people are likely to continue to suffer.
"Parents can help by having regular conversations with their children about what is and isn't OK online, and encourage them to seek help if they are being bullied, blackmailed or see anything that upsets them.
"It's important that young people know that if they need somebody to talk to, they can call ChildLine on 0800 1111."
Children's charity Action for Children welcomed the steps taken by Ask.fm today, saying they could not come "soon enough".
Matt Downie, head of campaigns and public affairs, said: "To have the ability to report abuse - which must be made clear - and have all cases looked into within a 24-hour window will help to address what young people are facing on a daily basis.
"We hope that the dedicated site Ask.fm has pledged to create will help parents understand how young people are communicating with each other, and that this becomes best practice for all social networking sites.
"It's important that parents talk to their children about their online experiences, to understand who they are contacting and what activities they are engaged in.
"As part of this they should continue to discuss what risks there are online and how they can be addressed."
Ask.fm said an updated function for reporting abuse would be live by September 2013 but that new moderation staff would not be in place until January 2014.
Ask.fm founders Ilja and Mark Terebin said they would create a new website for parents of its users to help them understand the site's functions and moderation policy.
Ask.fm said it was taking steps to encourage its users to sign-up using an email address to allow them to better track reports of bullying.
Ask.fm has announced that it will hire a safety officer to oversee the moderation of questions and comments on the social media site.
Part of a statement released by the site's founders, Ilja and Mark Terebin, said:
The father of Hannah Smith, the 14-year-old who committed suicide after being buillied online, has told Daybreak shutting down Ask.Fm would be "a waste of time".
However, David Smith felt Ask.Fm needed to be safer and raised concerns none of Hannah's online bullies had been brought forward by the website.
The website linked to the suicide of teenager Hannah Smith is due to announce the action it will take following a full and independent audit of its site and safety features by law firm Mishcon de Reya.
The 14-year-old from Lutterworth, Leicestershire, whose funeral was held on Friday, was said to have endured months of torment on controversial question-and-answer website Ask.fm before she was founded hanged in her bedroom two weeks ago.
Ask.fm delayed making an announcement based on the lawyers' recommendations on Friday as a mark of respect to Hannah.
Latest ITV News reports
The father of Hannah Smith, who committed suicide after being bullied on the Ask.fm website welcomes the site's new cybersafety measures.
Major organisations have withdrawn adverts on social networking site ask.fm following the suicide of teenager Hannah Smith.