How can I influence my parents to observe prayer on time, and alert them and my siblings to the importance of prayer? And should I follow up with them? And what is my mother's role in that?

You must exert your utmost effort to guide your family by supplicating for them, showing kindness to them, and utilizing all means to convey the truth and persuade them, while being careful not to confront them with what they dislike. Seek assistance from relatives, scholars, imams, and female callers to Islam who have an influence on them to help you in this. Suggest reading books on encouragement and deterrence at home, and change your approach if your father dislikes it. Try to convince your mother to help you in guiding the family.