I was wondering if teachers are allowed to be officers, and if that has to be included in our by-laws?

Unless it is prohibited by your bylaws, yes, teachers can be board members.

Sometimes there is no mention at all. Some organizations have restrictions. I belonged to one of each.

The one with restrictions had a very large board that included principal, vice principal and two teacher reps as well as all the officers and committee chair. Our restriction was that no campus employee (or district employee in a management position) could serve as one of the 5 elected officerpositions. They could still be on the board in one of the 20+ other roles, just not elected officers. My other PTO was wide open. Anyone could be anything.

Unless it is prohibited by your bylaws, yes, teachers can be officers and board members.

Sometimes there is no mention at all. Some organizations have restrictions. I belonged to one of each.

The one with restrictions had a very large board that included principal, vice principal and two teacher reps as well as all the officers and committee chair. Our restriction was that no campus employee (or district employee in a management position) could serve as one of the 5 elected officerpositions. They could still be on the board in one of the 20+ other roles, just not elected officers. My other PTO was wide open. Anyone could be anything.

With parental involvement almost non-existent at my son's school, I would certainly welcome anyone who wanted to be an officer or board member. Our by-laws don't state any restrictions regarding this matter. As a matter of fact, our PTO Treasurer is the school's 4th grade math teacher.

Our by-laws don't state, but we have an unwritten rule that the treasurer must be an employee, usually a certified teacher.
With the huge turn over in students/families, this is just a better/safer approach.