Is it permissible to pray behind a father who makes mistakes in the rules of Tajweed, and is the daughter sinful for not correcting him?

Your respect for your father and your concern for his peace of mind are commendable, but they should not prevent you from advising him gently and kindly. You could say to him, for example: "I heard a شيخ (sheikh) say such-and-such, so perhaps you could inquire about this issue." Advising him to attend congregational prayers in mosques is important because there is much good in it. As for your father's Tajweed error, if it is a manifest error (laḥn jalī) that changes the meaning or substitutes a letter in Al-Fatihah, then his prayer is invalid, and it is not permissible to pray behind him. If it is a hidden error (laḥn khafī) or a disregard for the recommended Tajweed rules, such as madd (lengthening) and ghunnah (nasalization), then his prayer is valid, and it is permissible to pray behind him. His stopping on a silent letter with a vowel mark does not invalidate the prayer.