There are no supplemental forms required from the parent to allow their son/daughter to travel with S-Trip! other than the S-Trip! Code.
As we are travelling as a group, the trip leader would have a copy of the registration and contact information for all travellers in their group and would provide this, upon request to the relevant border agency.
When there are issues of custody dispute and permission to travel is in doubt, S-Trip! would require a letter of permission to travel from the parent who currently has custody (or both parents when custody is shared). Generally this document would need to be carried by the traveler and would be asked from them during the time they go through customs and show their passport.