It's nice when everyone in the local community celebrates Eid on the same day. It'd probably be awkward if each parent did it on two different days. Hooray for unity, amirite?

There's more than one Eid throughout the year
Eid is a religious festival

there's a Major Feast and a Minor Feast

The former is when we do a sacrifice and give food to the poor
it reminds of Prophet Abraham's test with sacrificing his son

the latter comes after Ramadan, a month of fasting
it's not literally fasting the whole month
it's day to day

I hope I explained correctly