Yes, is there any specific reason for that "I" at the beggining? Whoud it be wrong to say "Miei genitori mangiano riso"?
Yes it's incorrect.
You need the article in front of possessives. Except in a couple of cases.
Why use il riso in the sentence Il suo cavallo mangia il riso but not use il in i miei genitori mnagiano riso?
Now that is a good question. A good, good question. I have an Italian friend who is now asleep; but I will ask her in the morning and will save this thread till tomorrow.
Enough of rice-eating parents! What we need are parents who drink prosecco by the bowlful.