Nikki Shaw, 32 and Dan Rowe, 36, from Market Harborough are being supported in their campaign by Kettering General Hospital after their son Ned died earlier this year.
Nikki said: "By telling Ned's story I want people to know how serious flu is. I want people to know how deadly it can be."
Ned was one of only three infants to die of flu during the 2018-19 season, but 2,924 people in total were admitted to intensive care units with flu and 1,692 people died.
KGH's medical director, professor Andrew Chilton said: "While most people recover from flu some will not. Younger people and older people – especially those with certain long-term conditions – can be especially vulnerable.
"Getting vaccinated is very important. The more people who are vaccinated the greater immunity is created within the community. This makes it harder for flu to spread and cause the damage that it does every year."
Ned and his twin Gus were born on December 7, 2018 at Leicester Royal Infirmary. When Ned was just six-weeks-old on January 24, 2019, Ned became poorly.
His mum Nikki was feeding Ned when he suddenly went cold, grey and floppy.
She said: "We acted quickly and dialled 999 and started CPR straight away.
"The paramedics arrived within minutes and we were rushed to Kettering General Hospital where the team managed to bring out baby back - but he was in a critical condition and had to be on a ventilator."
Little Ned needed a paediatric intensive care bed and had to be transferred to Addenbrooke's Hospital in Cambridgeshire later that afternoon because it was the closest available space.
A virus screening revealed the Ned had the flu virus but it was a rare complication.
Dan said: "The damage caused by the effects of the virus had been too much. There had been a very specific, but significant, effect to one part of his brain which controlled his breathing.
"He couldn't sustain his own breath."
Ned spent 12 days in Addenbrooke's on life support and with his parents by his side.
Nikki said: "When they diagnosed the complication we realised that our seven week old baby wan't ever going to get any better - and he wasn't going to come home to his brother."
On February 4, 2019, Ned's parents made the difficult decision to remove the ventilator that was maintaining his breathing.
"He managed to survive for a short while after extubation - but then he couldn't anymore. He passed away in our arms at 1.57pm in the afternoon," said Nikki.
Nikki had the flu vaccination while she was pregnant and Ned would have been protected from most common forms of flu, but he developed a less common strain.
However, flu can be deadly in all its forms and 11.7million people who had the jab in the 2018-19 season would have been protected.
Dan said: "How many times we had that conversation - it's just the flu. People underestimate the impact it can have and just how serious it can be."
Nikki added: "There is no reason why you shouldn't have a vaccination if you are able to have one.
"I will be encouraging every mum, in every play group we go to, to have their vaccination, to have their children vaccinated. Because the worst has happened to us.
"Ned's twin baby brother Gus is nine-and-a-half months old now. He is a beautiful reminder of everything that we could have had with Ned. He is our entire reason to be some days and I don't want to put him at risk.
"I am going to have him vaccinated as soon as we are able to."
Professor Andrew Chilton said: "On behalf of the trust I want to say how very grateful we are to Nikki and Dan for having the courage to share Ned's story to help prevent the spread of flu within our hospital, within our local community and across the country.
"I really hope that Ned's story helps to persuade a lot more people to have themselves and their families vaccinated."
KGH are aiming to honour Ned by having at least 80 per cent of their front line staff vaccinated, which is nearly 3,000 people.
Certain groups of people can get their flu vaccine on the NHS, including children aged 2-11, pregnant women, people aged over 65 and people with long-term health conditions.
More information can be found online here.