The Prophet (saw) said, ‘The first thing we will do on this day of ours, is to offer the (Eid) prayer and then return to make the sacrifice. Whoever does so, he acted according to our Sunnah…’ (Bukhari)
So, is it only a Sunnah and not a fard? The Hanafi school states that any adult, sane Muslim who possesses the minimum amount (the Nisab) must sacrifice and donate Qurbani. In short, if you are eligible to pay Zakat because you possess more than the Nisab, you need to offer a sacrifice (Qurbani or Udhiyah).
The Maliki and Hanbali schools state that the person responsible for the household can make the sacrifice or Udhiyah or Qurbani on behalf of the members in their immediate household. However, if you live in a household where more than one person possesses more than the Nisab amount and pays their own Zakat, then the best and safest option is for all those who pay Zakat to offer their own sacrifice or Udhiyah or Qurbani.