According to this you don’t have to if he is not listed on the birth certificate (for minors under 16). Down almost at the bottom of the page it states:
“One parent (with sole legal custody) must: Submit primary evidence of sole authority to apply for the child with one of the following:
Minor’s certified U.S. or foreign birth certificate listing only the applying parent.
Court order granting sole custody to the applying parent (unless child’s travel is restricted by that order)”.
So if you have a court order giving you sole custody you could use that as well, but the birth certificate should be enough. There are a few other options listed as well.