Paul Edwards, 48, and wife Belinda, 50, were on holiday at Flamingo Land, in Malton, North Yorkshire, in July 2010 when a family meal turned into a row about disability rights.
Mr Edwards is registered blind, the couple's daughter, Isla, 7, has cystic fibrosis and their other daughter, Melissa, 13, has Down's syndrome and autism.
The couple, from Kilmarnock, East Ayrshire, wanted to buy a meal from a site restaurant, but eat it in the open on picnic benches a few feet away.
They said they thought it would be too difficult to get the family inside because Melissa's disabilities make her prone to lash out and enclosed spaces can be very stressful for her.
However, Flamingo Land said that was against its rules and park staff refused to allow the severely disabled family to eat where they wanted.
Eventually, they were given their money back and asked to leave, but they sued the company that runs the park for disability discrimination.
Claims by Mr Edwards and Isla failed, but Melissa won a £4000 damages payout, which was then overturned by another judge, before now heading the Court of Appeal.
After Judge Lord Justice Pill said there were arguable grounds, the family's battle is now set to go to a full hearing before three senior judges on a date to be set.
Melissa's appeal case is being funded by the Equality and Human Rights Commission.