The Weekes have expressed their disgust that the couple have not yet served their sentence, and have repeatedly delayed the appeal process by not showing up in court.
"We are disappointed but not surprised that Mr Al Thani is refusing to attend court," a statement from the Weekes' said.
"We continue to hope that our three children will receive justice, but today we will be reflecting on the short time we had with them."
Last month, Prime Minister John Key raised the issue in Istanbul with the Emir of Qatar, Tamim bin Hamid Al Thani.
A spokeswoman from Mr Key's office said the governments would "stay in touch with each other to see the process through".
"The Emir assured the Prime Minister that he wanted to see justice done, that he is personally following the situation, and that he is committed to seeing justice for the family."
Having Mr Key's support was meaningful for the family, the statement said.