However, Health Minister Michelle O'Neill said no such plan was being considered.
Charles and Elma Hamilton's son, Ian, took his own life in 2013.  He was 38 and a father of two children.
He had suffered from mental health problems for many years.
An inquest heard that 24 hours before his death, Ian had been to three hospitals in two different health trusts.
Ian was first diagnosed with depression in 1998. He was deeply troubled and had problems with alcohol and drug addiction.
However, his post-mortem showed no sign of substance abuse.
His parents said the establishment of a single trust to deal with people with psychiatric conditions could help other people in the future.
The couple believe a single system would help ensure that a patient's notes, capturing their full history, would be accessible to all medics, no matter where they showed up for help.
Charles Hamilton said part of the problem for Ian was that he moved between health trusts.
"I think if they had a single mental health trust, where everything was collated together, all the information concerning the people presenting with mental health problems would be there there at the push of a button," he said.
"That way, professionals can access the patient's history and there are no delays in getting that patient treatment."
Elma Hamilton said Ian's final hours were very distressing.
"He went to three hospitals in two trusts inside 24 hours, trying to get help, pleading with people trying to get his point across," she said.
"He was frustrated and hopeless - he just couldn't get the answers he needed. In one hospital, he was waiting 10 hours to see someone from a mental health team.
"In a different hospital, he self-harmed. He was tired having to go through the same story so many times to different people."
His story highlights how, too often, there is a lack of joined up thinking between mental health services and how sometimes an emergency department is not the best place to treat those with a serious psychological disorder.
Their call for a single service across the trust is one of eight recommendations that was made in a recent report into mental health in Northern Ireland by the former Chief Executive of England NHS Lord Nigel Crisp .
"I think it is an argument - there is also an argument in keeping mental and physical health closely aligned," he said.
"I also understand in Northern Ireland there are lots of issues about people wanting local services and local control but even if you kept five trusts you could still run it as a single service, still have a single leader, the same standards.
"You could still have the facilities in five different places but run it across the entire system."
The family do not blame the northern or the south eastern health trusts but they are appealing to them to act on the findings of their son's inquest.
Elma added: "We don't want a stigma - depression is a health condition.
"But they need somewhere to go to feel safe and calm. Somewhere they will be listened to - not in an emergency department where doctors and nurses are busy dealing with broken legs and car accidents.
"You can't see mental health issues in a person. It is not evident but it is there."
However, Health Minister Michelle O'Neill told the BBC she was not considering a single mental health trust for Northern Ireland.
"I believe we can do better. We can make sure we are putting the services and the limited resources that we have to the best of fact," she said.
"I don't believe that setting up a a trust with all the red tape and everything that would go along with that is necessarily the best way to go. So I think we can do better."
The family of a man who took his own life have called for the establishment of a single mental health trust in Northern Ireland.