Yes, medical information can be provided to approved parties over the phone.
To be HIPAA-compliant, a phone call to a patient by a covered entity must specifically relate to a subject that the patient has previously given their consent to discuss. For example, a medical call center cannot cold call individuals and ask them about their health histories in the hopes of gaining a new patient.
A representative with a medical answering service can, however, take incoming calls from patients of a partner healthcare organization, access that patient’s files, and provide information to that patient regarding their previous visits.