Martin and Jane Weekes will make one final family journey to New Zealand next week.
The Weekes, who lost their triplets in a shopping mall fire in Doha, are bringing the bodies of Lillie, Jackson and Willsher home for a funeral.
Thousands are expected to attend a memorial service at St Paul's Cathedral in Wellington at 1pm on Friday.
This will be followed by a private burial.
A family spokesman said: "Martin and Jane have been overwhelmed by the support from people that they have never met, as well as friends and family.
"They would like anyone whose hearts have been touched by Lillie, Jackson and Willsher to come and share in that memorial service. They have been inundated with love and support." People are expected to travel from around the country to attend.
Martin Weekes told in an interview this week how he watched emergency services pull his daughter Lillie from the Villaggio Mall when fire swept through the complex in the Qatari capital of Doha on Monday.
Nineteen people died in the blaze - including 13 children.