He's about to start school. That depends on several factors. If he is and he shares joint legal custody with your partner, chances are he has the legal authority, along with your partner, to make medical authorizations. If the other parent isn't involved, your partner may list you as an emergency contact on the school's records, but she gets first dibs on medical treatment decisions. Regarding school permission slips, you may not have any standing here either.
Schools are legally responsible for the children in their care.
As such, they are only permitted to accept signatures from legal guardians. A legal guardian could be a foster parent or anyone who has legal custody of a child. As the partner of a legal parent, you probably don't qualify. You can also attend school events and conferences along with your partner, but chances are only she and any other legal guardian or parent will receive formal notification and be permitted to discuss the child with school officials. The best thing to do is ask your partner to contact the school to find out what she can do to maximize your rights with regard to school authorizations.