Is it permissible for my father to leave the Fajr prayer at its prescribed time and perform it before Dhuhr, given his struggle with urinary incontinence and his fear of urine leakage and its odor when praying it in the mosque?

Your father, who suffers from urinary incontinence, must perform ablution (Wudu') for each prayer and use a barrier to prevent urine leakage. Then, he should perform the obligatory prayer (Fard) and as many supererogatory prayers (Nawafil) as he wishes. He must be keen to attend congregational prayers (Salat al-Jama'ah) in the mosque, especially the Fajr prayer, while taking all precautions to prevent defiling the mosque. If he is unable to attend congregational prayer, he must pray Fajr at its prescribed time before sunrise, because intentionally delaying a prayer beyond its time is among the major sins.