Obviously, no one is supposed to forge a signature on a passport application or anything else.
Usually, however, criminal liability for forgery only arises when the fraud is material. In other words, if it is used to gain some significant benefit or cause some harm that wouldn't have been possible without the signature.
It is unlikely that this would be a crime, and unlikely that it would be prosecuted even if it was arguably crime, if the issuance of a passport for the child by the other parent for purposes of travel abroad had already been ratified by the other parent in writing or a text message, or if the other parent was required to agree to sign the application due to a divorce decree or separation agreement.
On the other hand, if the divorce decree or separation agreement was clear that an actual physical signature of the other parent had to be obtained and could be withheld by the other parent, and the communications between the parties were not sufficiently clear to constitute an agreement to sign the application, the forgery probably would be material.
Given that both parents had already signed one application and didn't dispute doing so, and that apparently both parents had agreed to allow the international travel that was contemplated with the initial application, it is hard to see how a forgery on the corrected application would be material. It might even be possible to submit the original application and a new application correcting the erroneous part of the original application together without forging a signature.
A previous answer concerning forgery in Australia is here.