Here is the pertinent part of the Ohio law:
": Ohio Revised Code §4301.69(B)
What If I Give My Child Permission to Drink Under My Supervision?
There are important exceptions to the laws discussed above. Under Ohio law, children under the age of 21 may drink alcoholic beverages while under the supervision of their parents. This means one parent (or legal guardian) must give consent and be physically present while the child consumes alcohol. Therefore, in addition to permitting supervised drinking in a private residence, Ohio law also allows parents to order alcoholic beverages for children at restaurants or bars – as long as the parents remain with the children while the drinks are consumed. Though it is legal for businesses to sell alcohol to parents (knowing the parents will give the drinks to their children), many restaurants choose to limit their liability by refusing to sell alcohol intended for underage children or limiting the hours that minors may be present on the premises.
Note that your physical presence as a parent or guardian is a strict requirement. You cannot “send” your child to a bar or private party with written or oral permission to drink. It is also not enough that other parents supervise your child while he or she drinks – you, as your child’s parent, must actually be there while alcohol is consumed.
So remember, when you have people over for the big game, you should monitor who has access to the beer. Yes, you can let your kids have a beer under your supervision. But if their friends sneak a few drinks, and their parents are not there to give consent, you may be responsible not only for permitting underage drinking, but also for any trouble or mischief they might get into later on in the night while under the influence."
In Ohio at least physical presence is VERY important. Like others have mentioned most likely everything would be fine. However, the consequences of something going wrong could be substantial.