Social workers raised a series of concerns about the way the boys, who are aged between nine and 14, were being cared for.
Judge Joanna Vincent has decided that the youngsters must live with foster carers.
She said their home was not clean, there were too many pets, they had missed school too many times and their parents could not provide the care they need.
The judge also said their mother was a “permissive” parent who struggled to say “no”.
Judge Vincent revealed detail of the case in a written ruling after analysing issues at a private Family Court hearing in Oxford.
She said the boys could not be named.