Can an officer give a minor a breathalyzer test without parents knowledge at school

The test is not invasive and children have fewer rights in school. My guess is that this is fine. My answer is you should talk to a criminal defense attorney who handles juvenile law and dui law, in your area.

How else to determine if the minor is driving with alcohol in their body, a blood test?

Yes, there is no duty to notify parents if the child is being investigated. The minor does have the right to refuse, but may be suspended. Assuming that your child had alcohol on her breath, you should consult an attorney in your county who handles school suspension and criminal cases.