To search for house sitting assignments without pets, just use the 'Pet type' dropdown on the house sitting assignments search page, and select 'No pets' from the dropdown menu.
Please note that generally we don't have that many listings for house sitting assignments without pets. However, not all of our home owners create 'Sitter wanted' listings on MindMyHouse. Instead, they can search for house sitters and contact them directly. Therefore, we don't fully know how many home owners without pets there actually is!