If I am considered the "domicile parent" can the other parent pick the children up whenever she wants other than the court ordered visitation days?

Exact use of the words vary by location, so where is this, in our state we use a term Primary custody parent if you have the custody of the child, and the other parent only visitation.

A parent with visitation may only get the child on their days and times, unless the custody parent agrees

This situation is taking place in Louisiana. The father is the domicile parent and the mother has visitation on every other weekend. The mother claims that she and the father has joint custody and she can pick.the children up whenever she wants.
Is this true?

This situation is taking place in Louisiana. The father is the domicile parent and the mother has visitation on every other weekend. The mother claims that she and the father has joint custody and she can pick.the children up whenever she wants.
is this true?

If the visitation is court ordered, then no this is not true. They may have Joint LEGAL custody, which means both parents have to decide on things like school and medical care.

But since the father was assigned as domicile or primary custodial parent, then they don't have joint physical custody and the mother is restricted to the court ordered visitation.

...The mother claims that she and the father has joint custody and she can pick.the children up whenever she wants.
...

What ScottGem said.

Ask her to point to language in the custody/ visitation order to back up her position. I suspect that she cannot.