Addressing the first national drugs conference for parents, Dr Mawhinney called on parents to become 'missionaries' taking the message back to their communities that drugs are damaging and dangerous.
Later, he departed from his scheduled speech when challenged by Esther Rantzen, who helped to set up ChildLine, to do something practical to help parents keep their children off drugs.
Ms Rantzen, presenter of the BBC's That's Life programme and a member of the Government-funded Health Education Authority, said parents were keen to understand more about drugs and to warn their children about the dangers, but they lacked the knowledge and training.
Admitting he was ignoring the usual rules about not making pledges 'on the hoof', Dr Mawhinney told a press conference: 'I am very prepared to look at this. If we are to focus on parents, we need to examine whether there is more we can do in training to supplement these occasional conferences.'
Ms Rantzen told the conference about one mother who, without any support or training, had a personal mission to visit schools to warn children about the dangers of drugs by telling them about the death of her son after he took amphetamines at a rave party to help him stay awake.
Offering to meet the mother, Dr Mawhinney said: 'I will be happy to look to see whether we can do some more structured things about training for parents and also look at whether the Health Education Authority can do more.'
Dr Mawhinney told the conference in London that parents had a crucial role to play in fighting drug abuse by young people. Some researchers claimed that children as young as eight were exposed to drugs, he said.
Quoting stark figures which showed that 122 young people died 'needlessly' in 1991 from solvent abuse, 40 of them experimenting for the first time, he said: 'Every parent must wake up to the fact that it could be their child lying dead in a mortuary from a drug overdose or solvent misuse.'