In case of a divorce in Germany, the parents have to reach an agreement on custody of the common underage child (s). Considering the well-being of the child, the family court takes a decision on custody of the child and gives the parents parental responsibility jointly or one parent the sole custody. (See also www.europa.eu/youreurope/citizens/family/children/parental-responsibility/index_en.htm)
The custody for children can be separated in several parts. There is personal custody, custody concerning financial issues and legal custody. You need the right for personal custody awarded from the court to make decision concerning the school education of a child. In Germany, it is common that the parents have a shared custody for their children. If only one parent has the custody, only he or she can exercise the right of custody.