How old are you? If you are over 16 (in the UK at least), they don't need to know about any trips you take to the docs, likewise they don't need to know about meds, or therapists (unless you are deemed very high risk to yourself, as in about to kill yourself).
Do you know why you worry? That can be conquered and dealt with so that you worry less.
It also sounds like your parents are ignorant. To be honest, they sound like they are scared of mental illnesses.
Maybe the worry and the stress about your parents reaction, etc, is making things worse? Making you worry more?
Is there someone you can talk to? Maybe a teacher? Or colleague? Or employer or friend? Who could help supprot you emotionally?