Students have a right to confidentiality in education, but this right does not extend to limit parent participation in education. Consequently, a parent isn't violating confidentiality when she observes a class, but she's limited in the information she can seek. She can't, for example, demand to see other students' grades, papers or medical records, and can't sit in on parent-teacher conferences for students who are not her own. If you're having a private discussing with a student at your desk, the parent may not be allowed to listen to this discussion, depending upon its contents.
Many school districts require that parents notify teachers prior to observing classrooms, but there's no specific law requiring notification. If a parent shows up unannounced, you may want to check school policies. In Bedford, Mass., for example, the Bedford Public School System requires that parents submit a formal request for an observation two days in advance, providing the names of the student and teacher, a signed release, a request for the time of day at which the parent wishes to observe and information about the purpose of the observation.
School districts are entitled to place reasonable limitations on observation times. For example, some school districts won't allow parents to observe classes during standardized testing. Bedford Public Schools requires that parents not share confidential student information they gain from their observations with any third party and mandates that observers not interact with teachers during class. Disruptive parents are generally prohibited, and if a parent becomes disruptive or combative, you may be able to request that she leave.
Teachers, like all individuals, have the right to be free of harassment, defamation and threats. A parent can't use her observation period to bother or threaten you, and can't make false negative statements about what she saw during the observation. If a parent is threatening you, you may be able to seek a restraining order, but should consult with the school administration first. Your school may adjust policies to deal specifically with a harassing parent.