Can the ratio of a parent's age to his or her child's age ever be exactly 10 : 9?

Mathematically, yes.

Realistically, no.

The difference in their ages is the parent's age when the child was born.

Let's say that the youngest possible is 12.

Then the ratio would be 10:9 when the parent is 120 and the child is 108.

Not likely.

Yes, if the parents had the child at a very young age. ex. Parents: 80 years old Child: 72 years old