If you are a private sector employee the answer is unfortunately NO!
However, by mutual consent between employer and employee, taking time off from work without pay for various excuses, especially for sick children, is always possible in practice. Another way to attend or take the child for medical examination, you can use some part of your annual vacation that can be divided by mutual consent, provided that one of the parts shall not be less than ten days.
In Turkish Labor Law unpaid leave is granted only to mothers and adoptive parents after childbirth and adoption. There are no special arrangements for parents of sick children, unless the child is disabled as explained at last part of the article.