Registering a child in his biological mother name

Dear Sir/Madam

I have a son who immediately after his birth was registered under my parents' name.

After I gave birth to my son, my parents suggested to registering him under their names. Since the biological father walked away from his responsibility of being a father, my parents convinced me that my son needs to be under their names to enable them to act as parents for him at school. In addition, they convinced me that my son will be insured under their names, under SSS GSIS and health care, which I accepted.

Now 10 years later. I got married and live abroad with my husband. Since my circumstances had changed, now I want my son to join me and his little brother in my new country, where I can provide day-to-day care for him.

Kindly, I urgently need an advice about how to transfer my son's name to my name legally without causing any legal problems to my parents.

I look forward to your reply.

Yours Sincerely
Eunice