Whether a student tells a school counselor anything real depends on what she believes will happen next. Most students have never been told what the actual rules are.
Confidentiality in a school is bounded
Counselors work under professional ethical codes that treat information as confidential, and simultaneously under state statutes and district policies that require disclosure in defined circumstances.
Those circumstances generally involve risk of serious harm to the student or another person, and suspected abuse or neglect, where counselors are mandated reporters under state law.
The exact triggers, who must be told and how quickly vary by state and are amended over time, so the district's published policy is the accurate description rather than any general account.
Students fill the gap with guesses
A student who assumes nothing is private says nothing, and the counselor concludes there is no problem, which is the most common failure of the arrangement.
A student who assumes everything is private and then sees a parent contacted experiences that as betrayal, and typically stops disclosing to any adult in the building.
Both outcomes are produced by silence about the rules, and both are avoided by a counselor stating the boundaries plainly in the first minutes of a first conversation.
Parental access is a separate question
Educational records are governed by federal law giving parents rights of access, and how counseling notes are treated within that framework depends on how they are kept and classified.
Some notes maintained as a counselor's personal memory aids are handled differently from records placed in a student's file, and the distinction is technical rather than intuitive.
Referral changes the framework again
A counselor may refer a student to services outside the school, where different confidentiality rules apply and where minors' rights to consent to certain care vary by state and by service.
That transition is where families most often encounter unexpected rules, and where asking the receiving clinician about their specific policy is more reliable than assuming continuity.
What a student can ask directly
Asking what would have to be said for a parent to be told, and asking what is written down and where it goes, are both reasonable and are usually answered straightforwardly.
A school counselor is also not a therapist in the clinical sense, and a student needing sustained treatment requires a mental health professional, which the counselor's role is often designed to identify rather than provide.