How can I preserve my faith and deal with embarrassment when my family members are late for prayer, especially when its time is close to expiring?

A Muslim must feel shame before Allah and adhere to their religion firmly, fearing no blame from anyone, while upholding good manners and showing respect to elders. If prayer time coincides with your father's presence, gently and kindly ask him to go to the mosque, reminding him of the virtues of congregational prayer. If you fear the prayer time will expire, remind him of the seriousness of that. If there is no fear of the time expiring, there is no harm in delaying the prayer until you are home, while still reminding him of the superiority of praying in the mosque. To help yourself adhere to your religion, remember death, Paradise, and Hellfire, reflect on Allah's Names and Attributes, and recall the reward for those who support His religion. Whoever guards the limits of Allah, Allah will guard him. Strive in supplication, struggle against yourself, hold yourself accountable constantly, remember the texts of the Quran and Sunnah, and seek beneficial knowledge.