If parents earn illicit money from bars and usury, will the prayers and supplications of their young children be accepted, especially given the presence of Hadiths indicating that the worship of one who earns money from unlawful sources is not accepted?

All praise is due to Allah.

It is religiously obligatory for the father to provide for his child, covering housing, clothing, food, and drink. It is permissible for the child to take his sustenance from his father's money, even if it is ill-gotten, without affecting his supplications. The child should observe the following: not to be excessive in taking from ill-gotten money, to strive for self-sufficiency through lawful earnings, and to advise his parents to repent from ill-gotten gains. And Allah knows best.