Is it permissible for us to offer a sacrifice on behalf of my mother and father from my mother's accumulated retirement fund, without the knowledge of my siblings, especially since we know that my mother is very forgetful and hesitant about her desire for the sacrifice?

If the mother possesses wealth that she can use for sacrifice, then the Udhiyah (sacrifice) is a confirmed Sunnah for her. It is not permissible to buy the sacrificial animal for her without her permission. If she grants permission, then it is permissible for you to buy it and slaughter it on her behalf. The dislike of the brothers does not matter as long as the mother is sane and performing the sacrifice does not harm her; she is in greater need of Allah's reward. You must strive to advise the brothers, especially those who abandon prayer, prioritize the mother's hereafterly welfare, and seek to earn reward for her in every possible way. As for sacrificing on behalf of the father from the mother's wealth, it is not permissible except with her consent and sound mind.