She also called up parents offering singing lessons for chil

She also called up parents是主句,主谓宾结构,描述她给其他家长打电话的动作
Offering singing lessons for children是目的状语从句,描述她打电话的目的是教孩子唱歌,offering描述伴随状态所以用现在分词,for children 是介宾结构.