I'm trying to find information for a family member. Parent A is dating a registered sex offender that is currently not allowed to be around child in question. The offender is taking classes that will allow this person to be around children and then moving into the residence is what we have been informed from parent A. If parent B does not want the child living with offender does parent B have any say in this. Custody at this time is full physical with parent A but joint legal.

No they have no say. They may go, to court and ask for the child visitation to be admended to prevent, or at the hearing of the sex offender, they may object

They have no say in who lives with Parent A. But if they feel that Parent A is endangering the child, they can apply to the court for a modification of the custody agreement.