Senator Anning used the Nazi-associated phrase"final solution"and called for a ban on Muslim immigration.
But KAP Queensland leader Robbie Katter said Ms Palaszczuk may have broken the law when she threatened to cut staff if the party did not condemn Senator Anning's remarks.
"The Premier should stand down during the time this is being investigated and the decision should be shelved while this is investigated,"Mr Katter said on Tuesday.
KAP said the Premier's move would undercut how well the party could represent the people who elected its members to Parliament.
Liberal National Party Leader Deb Frecklington,her deputy Tim Mander and LNP frontbencher Ann Leahy have also been referred to the Crime and Corruption Commission.
Mr Katter said he had referred the LNP members because they had been instrumental in forcing Ms Palaszczuk's hand by calling on her to act if he did not denounce Senator Anning's inaugural speech.
He was confident he was making the right choice in standing by Senator Anning,despite the fallout that decision has created for his party.