Bismillahir Rahmaanir Raheem
Al Jawaab Billahit-Tawfeeq

It is not just permissible but rather it is virtuous to remember Allah whilst doing chores. It is narrated by Imam Muslim that Aishah (Radiallahu Anha) said the messanger of Allah (Sallallahu Alaihi Wasallam) used to remember Allah all the time. Scholars clarify with regards to this Hadeeth that the messanger of Allah (Sallallahu Alaihi Wasallam) used to do Dhikr with tongue all the time except in two situations: whilst in the state of Janabah (impurity) and when entering the toilet.

وعن عائشة، قالت: كان النبي صلى الله عليه وسلم يذكر الله عز وجل علي كل أحيانه. رواه مسلم.

And Allah knows best