As a primary care provider's office, can we release copies of immunization records to a public school ?

Without signed authorization from the parent/legal guardian, can immunization records be released to the public shool system? Everything I've found in the HIPAA regulations says we can to a public health entity, social services, certain cirstances for law enforcement. But, I don't consider the school to be either of those - they're a public education entity, in my book, and public schools have procedures to notify a parent/guardian if they find a child's immunization record incomplete, then it's the parent/guardian responsibility to submit the info to the school system. I personally rcvd one of those notices for my daughter @ her 6th grade year. I would have been ticked if I'd found out my doctor's office had released any records without my consent. I wanted to be made aware 1st, if she needed updated imms. I had been told she was complete on imms until a certain year, but then the CDC added an extra booster which meant she did need another imm. The school's job is to notify the parent/guardian - not to obtain medical records. Am I incorrect?