Definitely not, unless they were ones the children had cosigned for or otherwise taken on. There are certainly debts that the parents estate may have to pay--for example, say that the parents owned a home worth $300k with a $200k mortgage. If the children inherit and sell the home, the mortgage has to be paid off before the children get their money.
So there are debts that the estate would have to pay and could reduce how much children might inherit. However, the children do not have to into their own pockets for any of their parents' debts.