Al-jawab billahi at-taufeeq (the answer with Allah's guidance)
Firstly, Yazid (Radhiallaahu Anhu) was the son of Hadhrat Mu’aawiyah (Radhiallaahu Anhu).
Secondly, death of Hussain (ra) has not been decisively established that Yazid himself killed or ordered the unfortunate killing of Husain (ra). There are some reports that he expressed his remorse on the actions of his associates, and even if he did, then murder and other sins do not necessitate Kufr.
Imam al-Ghazali (Allah have mercy on him) states that it is even impermissible to say that Yazid killed or ordered the killing of Sayyiduna Husain (ra) let alone curse him, as attributing a Muslim to a sin without decisive evidence is not permissible. (See: Sharh Bad al-Amali by Mulla Ali al-Qari, P. 123-125).
One should not curse Yazid, as the Shia curses. Rasullulah (Sallallahu Alaihi Wa salam) has said, ""Whosoever impersonates a nation (other than Islam) will be (resurrected) from them on the day of judgment". (Sunan Abu Dawud)
Therefore, It must be remarked here that this is not an issue on which one’s Iman depends, as there is no reward in cursing him, nor will one be asked on the day of Judgement as to what opinion one held about Yazid and his actions. This is a trivial matter, thus many scholars have advised to abstain from indulging and discussing the issue and concentrate on the more immediate and important aspects of Deen.
May Allah Almighty give us the true understanding of Deen, Ameen.