IMHO It is very irresponsible for parents to travel anywhere without the birth certificate (the only legal ID) of their child.

It is accepted proof of identity of the child and the parental/guardian relationship.

One example: if anything happens to the child and medical attention is needed, without a birth certificate to ascertain your legal connection and your authority to act on their behalf your child will be subjected to whatever the doctor/local authorities' treatment without your input.