I have never had an issue with certain positions being co-controlled. For example, I have a husband and wife team that effectively co-controlls the Treasurer position. By the book the husband is the Treasurer and the wife is the Asst. treasurer. I have seen many groups where roles such as Fundraising and Reading Programs are co-chaired.
In regards to officer positions it's common enough, but I've never seen it a requirement.
The idea that it has to be one parent and one staff member is something that I wouldn't like. As you indicated, there is too much that can become an issue.
Now if your group is working with a private school, then you might be stuck with what you have. But if public, and your group is a 501(C)3, non-profit, then you should be able to control yourselves. My best guess is that a Principal, some time ago, helped define this odd set up. Having mutiple Treasurers can work, but only if there is constant communication. My old Treasurer and Asst. Treasurer used to be best friends who worked with each other, no issues. My present Treas. and Asst. Treas. are married, no problems. But one parent and one staff member, it's hard for me to believe that there is appropriate communication going on there.
Having dual Presidents can work, but again you would need defined roles. But in this case, who controls the meetings? Who is responsible for what?
This doesn;t really make too much sense. Having a Staff Liason for communication purposes makes sense, and maybe that's what the staff President is, but if this is how your Bylaws are defined I think you need to make some changes.
There have been many discussions on these boards about how having a staff member in any of the officer positions can be a bad thing. Sometimes what is in the best interest of the Principal is not necessarily in the best interest of the PTO, meaning the children in relation to the purpose of your group. Often PTO's fill the gap, but there are certain lines that we usually don't like to cross. For example, providing funds to pay someone's sallary. It happens, but most of us frown upon this.
So if your group has a staff member running it, how can that person be impartial when their boss is the Principal? This is why most groups try not to have staff members as officers. Should they be involved, most definitely. Should they have a vote, most definitely. Should the Principal be a partner with the group? of course.
The way I would handle it is to e-mail the entire PTO, including the Principal and the staff members and let them know that you are forming a Bylaw committee to update your group's Bylaws over the summer. Ask for volunteers to be on the committee, and allow anyone who wishes to be on it. You should personally chair this committee. Then update your Bylaws with normal officer requirements/procedures, voting, etc. In regards to this next school year, you may just be stuck with the situation as is, but you can define the roles of each officer. For example, put it in the Bylaws that the Treasurer must inform the President of any transactions. This way, if one of your treasurers provides a check to someone for something, at least you'll be in the know. Next I would start to lean more on the parent officers. Try to use them as your primary officer. Certain officer positions may not be an issue in regards to having a staff member. For example, Secretary. But president, V.P. and even Treasurer are all positions that can become an issue with a staff member in them.
Hope this helps,
PresidentJim