If the child has no income, then the answer is “absolutely not!” If the child has no income, but the parents have decided that he or she should partially pay for living in the parents' home, then please also pay the child for, for example, his or her housework—cleaning the apartment, taking out the trash, walking the dog, washing dishes, and other work that the child can do.
If the child already earns money but lives with their parents, then it would be logical for the child, as a full member of the family who consumes services, to pay for them.
The parent responsibility for the children by giving them residence not matter has no income or not because the parent responsibility prepare all before planning to have children. Just simple residence enough depend the parent economic condition and pay education cost exactly until how parents' financial capabilities. If later the children has own income if come from good family the parent will not ask to pay resident cost and want to see his children get stable with financial.
For housework like cleaning, washing and other kids of house work I think for children need to do it to help their parent maybe lack time due over working every day. When I am still children around 12 to 15 years old spent my time helping my parent from cleaning the house, washing until helping them in the farm.
I have a slightly different view. Parents are responsible for their children, that's a fact. But parents are not their children's providers. Parents' job is to raise their children, educate them, help them in the early stages, and teach them to be independent! Otherwise, they will grow up to be dependent.
Regarding payment for housework, I mentioned it as an option in cases where parents “demand payment” from a child who has no job and/or income. In such cases, there should be some balance—if you demand payment before the child becomes independent, then give them the opportunity to earn money at home.