Officials at the BT Murrayfield Stadium in Edinburgh said younger fans will have to be under adult supervision as there is a bar.
Those under the age of 12 are not allowed to stand on the pitch, while those under 16 will also need to be accompanied by an adult.
But passes for Tuesday's gig only specify supervision for under-12s. The concert has been sold out with 50,000 tickets and parents now fear that hordes of children could be left standing outside.
The restrictions were revealed as the previously squeaky clean image of the band has been rocked by a video showing them apparently smoking cannabis.
Louis Tomlinson was at the centre of fresh controversy yesterday over claims that he apparently used a shortened version of the n-word in the same video.
Laura Barfoot, who is taking her six-year-old daughter to the One Direction concert, said: "It wasn't advertised well. I have heard of people having to sell standing tickets and buy seated ones at a ridiculous price because they still want to go but can't take their child because there's a restriction."
Councillor Jeremy Balfour has demanded answers from operators over claims restrictions were not properly advertised by ticketing websites or the concert promoter.
Mr Balfour, who is on Edinburgh's licensing board and represents the Corstorphine/Murrayfield Ward, said: "I have very serious concerns about the event and the safety of the young people involved, if they are left unaccompanied outside the concert for several hours.
"If anyone under the age of 16 is unable to enter the event they should be given a full refund and an apology."
Organisers insisted they would sell alcohol to adults and that they had assessed the risks involved.
Promoters said the terms were clearly stated when tickets were bought and insisted no refunds would be issued. But ticketing website Ticketmaster, recommended by tour promoters, allowed buyers to get to payment without reading about age restrictions.
A spokeswoman for the Scottish Rugby Union, which owns Murrayfield, said: "We are selling alcohol because there is a proportion of the audience who will be of the legal age to consume alcohol. We have assessed the risks and put the appropriate measures in place to ensure this is managed correctly.
"Anything related to ticket sales and conditions should be directed to the relevant ticket sales agent. At the point of sale on these official ticketing websites, all these conditions are prominently displayed."
A spokeswoman for DF Concerts, the gig promoter, said no refunds would be issued, adding: "The age restriction has been communicated via our event website, at point of sale and on the Ticketmaster website.
"We have also sent a direct e-shot to ticket buyers. Plus, a range of communications will be issued in advance of the event."
A spokeswoman for Police Scotland said: "Police are working with our partners at DF Concerts, Murrayfield Stadium and the city council to ensure a safe and secure event."