Should exes who share children together have a say in each other’s partners?

So let me explain. You are with someone and you share a child / children together. You break up.

One of you moves out and the child will spend 50% of their time with the other parent in their home.

They meet someone else and they decide they want their new partner to move in.

Meaning that the time spent at the other parent’s house, their new step parent is there too.

Do you think in that scenario the other parent should be entitled to a say in who is about to become their child’s step parent?

I know that if my child was spending 50% of their time with this new person who is going to become a parental figure, I wouldn’t be entirely comfortable until I know what kind of person they are.