I'm adopting through the foster care system. The mother relinquished her parental rights and the father wants an open adoption, but we do not want an open adoption. Do we have to give the parents any rights ?

A judge might order a contact (open adoption) agreement over your objections, but probably not.

Although I don't know the particular circumstances of why the children were removed, you should think carefully about what is in the best interests of the children long-term. Children love their parents even if their parents neglected them or used drugs. They often long to know that their parents love and care about them in return. Allowing a few supervised visits a year from a birthparent would give them great peace of mind. This is true even for children who are happy in their adoptive homes.

Even if you don't really like the birthfather, and whether or not you do allow contact, it is very important to always speak respectively about their birthparents. Children who hear negative things about their birthparents
tend to internalize the message and come to believe that something is wrong with them. This does not mean you have to lie about neglect, for example, but you should explain it in a way that is respectful. For example, you always want to explain that the birthparents loved them, but that because of problems they were having with drugs, for example, they could not care for them properly, which is why you adopted them.