Typically, there is a picnic on Hospital Point sponsored by the Alumni Association. There's s nominal charge and the food reportedly isn't exceptional (typical picnic fare) but they usually do have freebies from "vendors" such as USAA and it can be nice to meet some of the other parents.
You need to decide whether to spend the day on the Yard or return to your hotel for the day and come back for the Oath. I'd say that parents who: (1) are grads themselves, (2) live in the area, and/or (3) had another kid go through I-Day in years past, generally spend the day doing something else. This is because they don't really need to do all the stuff there is to do on I-Day. That said, I'm sure there are parents in the above categories who stay on the Yard for the convenience of not having to come and go and/or their hotel is a good distance away.
Those who aren't familiar with USNA and/or live far away so things like the MidStore are a treat may well enjoy spending the day on the Yard. You can also walk out in town but, as noted above, it can be REALLY hot so be prepared.
What you won't be able to do is see your child between the time you deposit him/her at Alumni Hall and after the Oath ceremony. [You can hang out at Bancroft for a brief glance but really isn't worth it]. So, you'll have 10+ hours to occupy your time.