An example of an organizing document would be the articles of incorporation or perhaps the bylaws if you do not get incorporated. An LLC is a type of corporate structure, not an organizing document.
The person who incorporates the organization doesn't necessarily have to be an officer of the PTO. It can really be anyone. Once the corporation is set up, officers will be elected to run it anyway.
What JHB is talking about is more the "registered agent" that a corporation needs to have. That's a person on file with the state who will agree to get formal notices about the corporation. You can pay a service to be your registered agent, or you can pick someone who will be around for awhile, like the principal.
Some set the principal as the "agent of record" because he/she tends to be around several years. Otherwise, you have to update that information every year and there may be an administrative fee plus the hassle of remembering.
Another option would be a teacher who is involved in PTO (and planning to be around) or an involved parent with a young child and no plans to move. No guarantees, of course, but if you have a parent with a kindergartner and a baby, that person will be involved in the school for the next 10+ years unless they move.
Use the school's address; never a home. It's just a contact point, so do what ever gives you the most continuity.
I have been reading up because we too are looking at incorporating, getting our EIN number, and filing for the 501c3. Everything that I have seen for our state requires a main person to be listed on the incorporation paperwork (we are in Missouri) and I don't know that I will be the president of the group next year.
Before you file for exemption, you need to get your corporate structure together. I'm not aware of any parent groups who set themselves up as LLCs--seems unnecessarily complicated. I've usually seen them set up as corporations or unincorporated associations. Your structure is regulated by your state government, so you might want to check with them--there's a list here: www.isquare.com/states/stateoffices.cfm. For example, I'm in Virginia and in Virginia there's technically no such thing as a "nonprofit"--we're referred to as "nonstock corporations." But what works for me might not fly in your state.
Our PTO is looking at filing the 501(c)(3) for the first time. Which is better to use as an organizing document--Articles of Incorporation or LLC? We are new to this and assuming there must be a difference......
