I am not sure I understand, no Board Member of a non-profit theatre should be taking any compensation for the work.

I think that the parents would probably be involved in the productions no matter what, at least that is what we find with out Youth productions, there are always parents ready to get to work.  Therefore they should not also make up your Board of Directors.  It sounds lke you need to start making the transition from a working board to a governance board.   The Board can, in fact should, still have parents that are involved intimately with the shows, but it should also represent the greater community of your supporters.  Look through your audiences and donors, look for business and community leaders. 

As for Board vs Paid staff, there is no "vs", well at least there shouldn't be.  They each have different roles to play in a healthy organization.  The Board should oversee finances, indeed they are liable, and also strategic planning, organizational mission, etc.  The staff should busy themselves on an organizational level, programs, advertising, operations in general.  There is no reason that you should need that person to be an Artistic Director, many healthy community theatres do not have one.  Finding a person to run the back end of stuff should be easy, finding someone to do it with the pay level it sounds like you can afford may be harder.

The eventual "administrator" could very easily be a current parent, intimate understanding of the program is vital, so not a bad starting place.  I would avoid the trading tuition model and make it a regular paid situation.  Employing someone though does bring some new oversight issues though that you need to be aware of.  Your current directors, choreographers, etc are contract personnel, the administrator should be an employee.  Check with your companies lawyer, or one who is a potential Board member if you don't currently have one, on the exact laws in your state as there are some differences. 

Good luck