As to the main question - it's a Private school - they have their own rules and don't need to have a parent on their board. To the larger issue...most public school school board meetings have a general session where there is opportunity to bring issues before the board. As a citizen it is your right to bring anything before the board. (Be aware that there are different practices for writing down questions, points of information, etc. and having them presented prior to your speaking - so check them out.)

Should an elected PTA/PTO president sit on the school board - nope, not unless they are elected by the general public.

Anyone may run for school boards, providing they meet the residence requirements set forth in board policy. One school's PTA president in our district was duly ELECTED to the school board, and is an incredible asset to that position with her experience, knowledge, and dedication. She also serves on the district accountability committee and two school advisory councils, among other activities. School board meetings are open to the public (except for executive session), and parents should be encouraged to attend and sign up to speak, as well become involved in their building SAC and DAC, which report directly to the school board and impact accreditation.

I think you might have more than one way to go on this. The school board is required to have open meetings - this means that anyone may attend. If someone is not on the board's agenda, they may not have time available to speak - this would depend on the board, how much time they have for their meeting and what all they have to accomplish. There is usually "visitor" time alloted in an agenda that will allow comments from the audience.

As a board member for our school, I have chosen to not allow myself to be an office for our parent club to avoid any possible conflict of interest either perceived or actual. Not everyone is able to keep both jobs separate. I am an active member of our parent club and do chair events or committees.

The school's teachers, staff & administrator have regular meetings/staff training that neither the school board nor the parent club attend. A representative from the board or parent club may attend at invitation from the administrator to give more details on an activity or situation. The administrator brings information from these meetings to the school board and a teacher representative brings the same info to the parent club.

I am a PTO president I try to keep myself updated on whats going by at least reviewing board minutes and keeping in touch with one of the Board members. I can't attend the meeting because of schedule conflicts, but I pretty much always know whats going on.
I think its very important to know not just PTO issues, but other things going on at the school so when parents ask you have an answer or can at least direct them.
There's nothing wrong with having more knowledge.

We also have our PFA President sit on the School Board. She holds a non-voting position. However, this allows her to give update on what we are doing and make sure that our goals are inline with the goals of the school in general. Why wouldn't a Board want a representative from the PFA to participate...aren't we all in it together?

We are a PTO and our principal and our teacher representative bring their ideas to the table from their leadership meetings. Our officers do not attend the Principal's meetings though she is an active member on our board. She and our teacher representative bring the necessary information back to the PTO to assist us in being on track.

We are suggessting a powerpoint type presentation that represents all the functions of the PTO. We have someone in business who is willing to create this for the school and the PTO. This might be the best way to show all what is involved with your organization. I am not versed in the PTA guidelines about media, so you may need to check with other sources for help.