Can a teacher sue a parent for slander?

Did the parent make any statement of fact which was derogatory and which she knew to be untrue?  Did that statement or statements damage you?  I can't tell from your question exactly what the parent said, but if she stated a fact (i.e. "this teacher is only 22 and has never traveled abroad"), and her opinion (i.e. "the teacher is too young and inexperienced to chaperone our children"), she has not defamed you.  If, on the other hand, she made a representation of fact which she knew to be false (e.g. "I have seen the teacher asleep in the classroom on more than one occasion"), she may have defamed you.  On top of that, in order to be actionable, any defamatory statement has to either be of a type which is considered slander per se (one type would be a statement which disparages your ability to do your job, so this may qualify) or you have to be able to show that the statement or statements caused you actual monetary damages (for example, if the defamatory statements caused the school to fire you.)