Their Starship journey began when Elliott was five weeks old. Two midwives in the family noticed her poo was abnormally pale, which led to a medical check.
"Two days later, we were at Starship and she was diagnosed with biliary atresia," Mrs Gulliver recalls.
Because of the rare condition, bile wasn't draining from her liver, which can harm the vital organ.
It has required three big operations and several smaller procedures including liver biopsies, mostly at Starship theatres in Auckland.
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The first operation was to enable bile to drain into the bowel. This can be a long-term fix, but for Elliott it lasted only two months, after which she went on the waiting list for a liver transplant.
A suitable match from a deceased donor came up after 21 days and Elliott, aged eight months, had the operation at New Zealand's sole liver transplant unit, at Auckland City Hospital.
The donated liver was slightly too large, so Elliott's cut tummy muscles had to be left open to provide enough space.
Transplanted livers gradually adapt to the size of the recipient and as Elliott has grown, her new organ has come to fit properly; in July she had surgery to stitch the muscles back together.
Mrs Gulliver said the gastro-enterologist and transplant surgeon who saw Elliott on Monday reported she was "doing fantastic".
"Her liver functions have been abnormal for 18 months and they have now just come into the normal range."
Because of the transplant, Elliott must take drugs to suppress her immune system.
She made two trips to Auckland because of "slight rejections of the new liver but it's been a pretty smooth ride since then", Mrs Gulliver said.
Watch: Lucy Lawless: Get behind Help Our Kids
Elliott's parents try to let their "chatty, energetic" daughter live a normal life - she goes to an early childhood centre - but because the immune suppression drugs leave her vulnerable to viruses and bacteria, they must be careful to avoid her being exposed to people unwell with infections.
Mrs Gulliver said Starship and Ronald McDonald House were a second home to her family.
"It's always exciting to see the nurses and the liver team. They have looked after our baby since she was five weeks old."
She urges people to give to the Help Our Kids campaign because "you never know when you or someone in your family or someone you know is going to need that service".
The campaign, which started in September to raise money for equipment in the new and upgraded operating theatres, will end on Friday.
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