What is the best way to deal with a sister who does not observe the hijab and whose behavior is displeasing, given that the parents refuse to intervene in her upbringing or provide her with financial support that might be spent on improper things, and is one held accountable for his sister's actions for not forcibly preventing her?

As long as you have enjoined good upon your sister and forbidden evil to her, there is no sin upon you. You are not to compel her as long as her parents are present. Continue to advise her with wisdom and good counsel, not with harshness or violence, for Allah Almighty says to Moses and Aaron, when they addressed Pharaoh: "And speak to him with gentle speech that perhaps he may be reminded or fear [Allah]."

As for giving money to your mother for your sister, if your sister will certainly spend it on something impermissible, then it is not permissible for you to give it, as it involves assisting in sin, and obedience to the mother is not obligatory if she commands a sin. However, if she will use it for permissible things, then there is no harm in you giving the money to your mother, out of piety towards her and to please her. We advise you not to withhold from your mother what she asks if you are able, but clarify to her that this is conditioned upon it not being spent on forbidden matters. By doing so, you clear yourself of responsibility and please your mother.