All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
An engagement is a promise between the two parties and each party has the right to invalidate it whenever they wish to do so. However, one should fulfill the engagement if there is no sound reason for invalidating it. Fulfilling a promise is desirable according to the view of the majority of the scholars and it is not obligatory. For more benefit on the ruling of fulfilling one’s promise, please refer to Fatwa 90273.
However, your parents (either or both) have no right to call off the engagement without your consent. Also, they should not prevent you from marrying this girl and you did well by trying to convince them to accept. You should not despair of doing so (trying to convince them) and you should first seek the Help of Allaah and then the help of whomever may influence them. Then, if they agree, praise be to Allaah, otherwise we advise you not to disagree with them or to do what angers them. This is provided that your heart is not attached to your cousin and you do not fear to be tempted by her if you do not marry her, otherwise, in such a case, it becomes permissible for you to marry her even if your parents do not agree; for more benefit, please refer to Fatwa 106420.
Finally, if marriage with her is not facilitated for any reason, then you should cut any relationship with her as she is a non-Mahram to you and it is not permissible for a Muslim man to have an affectionate relationship with a non-Mahram woman as we clarified in Fatwa 81356.
Allaah Knows best.