Is it permissible to perform tayammum instead of ghusl to cleanse oneself from janabah due to an inability to use water for a non-medical reason, such as fear of parental anger, which could lead to the loss of something important?

As long as you find water and are able to use it, it is not permissible for you to forgo it for tayammum (definition). This is because your family's anger, or your unwillingness to be embarrassed in front of them, or shyness are not valid excuses that permit transitioning from using water to tayammum. The jurists have mentioned the conditions for transitioning from ghusl (definition) to tayammum, and what you mentioned is not among them.