Hi clieslie,
The short answer is yes -- if your PTO wants to be an independent group. The school's status can't be "shared,'' so to speak. Many groups establish themselves as an independent parent group and then set the process in motion to become a 501(c)(3).
Here's a basic look at what it means to operate as a 501(c)(3)
what it means to operate as a 501(c)(3): https:/classic.ptotoday.com//pto-today-articles/article/394-501c3-for-ptos