Is it permissible to abandon the home due to the sister's open display of her adornments (tabarruj) and the parents' failure to command her to wear the hijab, especially given their deteriorating health when discussing the matter?

If abandoning the home means abandoning one's parents, then it is not permissible, due to the potential harm it may lead to. However, if abandoning the home means abandoning one's sister, then it is permissible if a benefit is hoped for. It is recommended to advise her gently and kindly, clarifying the seriousness of displaying oneself (تبرج) and mixing freely (اختلاط), and mentioning the texts that prohibit this.