Both parents, most times first name by the father, the mum gives the second.
I believe this is because the engagement act is done by one party in the relationship(the man), unlike in a wedding scenario where the priest officiates and both bride and groom gets to wear a ring.
Logically, the man should wear the ring but here logic sounds stupid because it’s a societal norm for ladies to wear a ring with a jewel on it on their middle finger to signify engagement, men neither have a type of ring nor a specific finger that signifies engagement; seeing there is no such norms for men,one would look really silly wearing a so called engagement ring because just you and your fiancee would know what the ring is for and that clearly flaws the ideology of engagement rings which is a clear statement, both for the couple and the public.
I think it’s just tradition, that the man proposes to the woman and gives her a ring. He can also buy a ring for himself to show that he is engaged. I know of a man who said he will wear an engagement ring when he is engaged.