If parents are divorced, is it acceptable for a counselor to allow an ex-spouse to be present during a child’s session?

I see a few issues here:

First, the age of the child is important.  If the child is considered an "adult" by state law with regard to counseling (states differ on this - the range is typically between age 13 and age 18), no one is allowed to be in the session without the child's consent.

Second, ethical and legal standards generally require that a counselor be in the role of a therapist, or in the role of an evaluator, but not both.  This means that a counselor should not generally provide both counseling and offer an opinion regarding who should have custody of a child.

Third, knowing the details of any parenting plan or separation agreement is important.  Generally, either parent can consent to counseling for a minor child and can be present during the child’s sessions.  If a court order, parenting plan, or separation agreement specifies that one of the parents has sole decision making authority, then only that parent can consent to counseling for the minor child and only that parent can be present during the child’s sessions.

In a situation like this I would strongly recommend seeking out legal advice specific to your state law.