Carter Cookson, born prematurely on Boxing Day, may only have days to live and a heart transplant is his only hope.
Sarah and Chris Cookson, from South Shields, South Tyneside, said they were focusing on spending time with him, reading him stories and “fussing” him at the Freeman Hospital, Newcastle.
The couple have already been through the heartbreak of losing their first son Charlie, who died aged two in 2013, which led them to set up a charity.
Their plea for an organ donor for Carter has been shared widely on social media.
Writing late on Friday night on Facebook, Mr and Mrs Cookson said Carter was “still with is, he is still fighting”.
Their post read: “We are spending as much time as we can with our very brave, amazing baby boy...
“We have done everything we can... we have reached out and beared our souls and heart ache... for him... to give him his best chance of survival.
“Our hearts are breaking... as once again no phone call... and another day closer to losing our baby.
“We are now focusing on being his Mammy and Daddy... reading him stories, fussing him, cleaning him... holding his hands and telling him how much we Love him xxx xxx
“Still praying... we will never give up until his last breath.”
The couple also thanked the public for sharing their plea for a heart donor.
Carter underwent surgery a week after his birth, but the operation to drain one side of his heart and install a pacemaker was not as successful as hoped.
He is now being kept alive with the help of an extracorporeal membrane oxygenation (ECMO) machine, which is pumping blood around his body.
After their first son died they set up the Charlie Cookson Foundation in a bid to support other families with seriously ill children.
Family friend Angela Moore has launched a GoFundMe page in a bid to raise money in order to support the couple, and has so far raised almost £2,000.
She said: “Since their older son Charlie gained his angel wings, they have dedicated their lives to raising much-needed funds to help families of children with life-limiting conditions.
“But now I’m appealing for any other families out there to step forward and help them.
“If there are any parents out there grieving the loss of their own child, please consider organ donation so you could possibly save another child’s life.”