The trio are recovering but Valentina has been left with respiratory issues from a lower lung infection, and Manaia still has rash scarring.
All three children were hospitalised and Peeni has said on Instagram, where she has 25,000 followers, she thought she and her husband would lose Valentina at one point.
"Measles is no f***ing joke. My babies are perfectly healthy and look at what it did to them.
"So imagine a child who has a weak immune system, or a heart condition or an adult who is fighting cancer.
"Measles can kill so don't be ignorant, be educated, be vigilant."
Both Carter and Valentina had received a single dose of the measles vaccine, which gives 95 per cent immunity, while Manaia was too young. The vaccine is usually given at 15 months and again at age 4.
Peeni said she and her husband and their eldest child, 6-year-old Marlee, were all fully vaccinated and none of them caught the virus, which has now infected 937 people around the country.
Originally from Australia's Gold Coast, Peeni said it was not her place to tell parents to vaccinate their children, but she urged them to watch for symptoms of the deadly virus to help contain the spread.
"Whether you're vaccinated or not, just look out for the signs because at the beginning it can just look like a common cold.
"It's a runny nose, it's a cough, it's fevers. Don't go anywhere if your child is sick.
"I cannot stress enough, I'm begging parents to please just know the symptoms. If you have that in the back of your mind you can help contain the spread."
The 31-year-old said a friend who had a child under 1 with a heart condition was confined to her home to protect the youngster from exposure.
Peeni decided to share the family's ordeal online so that other parents could see a real experience of measles.