Your by-laws should state clearly if you are to have co-presidents or a pres and vice pres. That is your out if your by-laws don't list co-presidents. You have no right to change your bylaws. Explain that to her.
I was ready to comment, but JHB sums it up so nicely from her personal experience. Any group, including a PTO, benefits from having one leader at the top: one person who takes the heat and responsibility. One person who has the final say, makes decisions, and deals with the results. If there's enough work for two people to share, then there's two jobs. Our PTO allows sharing of subordinate offices like vice president and secretary, but our bylaws clearly state there can be "one and only one president".
I think you are right to avoid this scenario - especially since it's not a situation you willing embraced.
Some here on the Boards have seen very successful co-presidents. My experience was the opposite. It was at an agressive time in our growth, right after an embezzlement. We were implementing strict procedures, filing for 501(c)(3), formalizing processes and moving forward with all programs. No one was too thrilled to be pres so a friend and I agreed to do it together.
We were both intelligent, dedicated moms. Good business backgrounds. What could go wrong? Split the work; half the effort.
Ugh! It was a disaster. There were two many decisions where someone needed to act on the fly. (Not the big voting issues, just endless day to day decisions.) It was impossible to decide everything together so one or the other was often left out of the loop. The friction increased until it almost cost our friendship.
It's not a scenario I'd enter into lightly again.
Absolutely talk to the principal. Recommend this person for another role. (Not VP, sounds like she'd drive you crazy. Something she can "own".)
