If both parents are traveling with a child, then I think there's an honor system applied to the statement that you are the legal parents. If just one parent, a grandparent, or a Girl Scout leader is traveling abroad with a child, then they're recommended to carry a letter documenting the other parent/s' permission. That's not really proof either, but it would at least be official-looking (notarized) paperwork stating your relationship.
If that's you, this is what our tour department suggests for tour members traveling with grandchildren, or single parents, etc.:
In addition to his or her passport, any child under age 18 who is not accompanied by both parents must bring a notarized letter, signed by both parents, allowing the child to be taken out of the US and also allowing the adult traveling with the child to seek any necessary medical care. The letter should include the child's legal name and date of birth as shown on his or her passport, plus dates and countries of travel. For details, call the US State Department at 888-407-4747.