Gatekeeper parent
A gatekeeper parent is a term sometimes utilized in the legal arena to refer to a parent who appoints themself the power to decide what relationship is acceptable between the other parent and the child(ren). The term is broad and may refer to power dynamics within a marriage or may describe the behaviors of divorced or never married parents.
Characteristics
Most "gatekeeping" situations are studied with consenting married couples who are first-time parents. Parenting situation studies using divorced couples and out-of-wedlock parenting relationships that show very similar or identical behavioral characteristics as married couples with children are usually studied as Parental Interference, Parental Alienation, Maternal Alienation, and Abuse by Proxy.
A gatekeeper parent exhibits the following behaviors:
Criticizes the way other parent, spouse, or ex-spouse parents
Creates unbending or unrealistic standards in order for the other parent to spend time with the children
Demeans or undermines the other parent's efforts at being an authority figure in the child(ren's) lives