You're on the right track by adding committee chairpersons. As you continue to get more parents willing to step up, in time, you'll fill those executive officer positions.
As our parent involvement increased at our school, so did our committees and the number on our executive board. We currently have one president, 5 vice presidents, recording secretary, corresponding secretary, two co-treasurers. Our vice presidents are assigned to different duties, such as, 1st vp is the assistant to the president or pres elect, 2nd vp oversees Membership (there is also a membership chairperson who runs membership), 3rd vp oversees events (basically is the go to person for all committee chairs), 4th vp oversees fundraising (there is also a fundraising chairperson), 5th vp oversees PR and communications (plans brochures, posters, media communications, press releases). One secretary handles all correspondence, the other handles minutes. One treasurer keeps the books and writes the checks (treasurer and president have to both sign every check for security), and the other treasurer balances the books (basically, is our monthly auditor) and puts together the spreadsheets for monthly meetings.
Our motto is, "if they want to volunteer, create a place for them." We laugh every year that eventually we are going to have 20 officers at the rate we are going...but I guess that's not a bad thing.

We have a small board and rely on committee chairs to do all the rest of the fundraisers and service projects. We have a president, president-elect (who spends the year learning the ropes), treasurer, secretary, and ways and means chair. The Ways and Means chair oversees all the fundraising events. It works great -

I am going to try to do the same thing. I will make myself as available as possible to these committees. There is too much to do at this newly formed school. Now just crossing me fingers we can get people to sign up for them.

I will probably list the committees and their functions to add to the info packets we will be giving out.

I agree with a tie-breaker position. And I think Secretary is the best bet. What about a Volunteer Coordinator to help organize all of the events?

I would add another board member. That way there could be a tie breaker if needed.

I personally feel committees are the best way to do things. That way people aren't just assuming some event is covered when no one has done anything or several people have done the same thing. I would definately keep a secretary as well as treasurer.