How can a mother be convinced to correct her prayer, given that she does not complete the obligatory glorifications? Is the prayer of one who prays like this invalid and must it be repeated? And must the father repeat his prayers if he separates the verses of Al-Fatiha by raising for bowing?

Your mother's prayer is valid. The obligatory glorification (tasbih) in bowing (ruku') and prostration (sujud) is once, while three times is recommended (mustahabb). As long as she has settled in prostration and glorified once, she has fulfilled the obligation. Your father's prayer is also valid, because raising the hands is a recommended (sunnah mustahabbah) act that does not invalidate the prayer by omitting it or performing it before finishing Al-Fatihah. The important thing is that he completes Al-Fatihah while standing. You should advise them gently and wisely. If they respond, that is good; otherwise, you have fulfilled your duty, and their prayers are valid.