After parties separate, it is highly likely that they will date other people at some point in the future. When that happens, then another person is pulled into the orbit of the parent's custody arrangement. This can be a emotional and contentious proposition, especially when children are involved.
In most Standard Visitation Orders, there are General Terms and Conditions that say who can be present when a child is picked up or dropped off. This language usually says, "either parent may designate a competent adult to pickup and return the child, as applicable; a parent or a designated competent adult shall be present when the child is picked up or returned" See Texas Family Code 153.316(6).
This means that if new boyfriend or new girlfriend, new husband or new wife, is there for pickup or drop off that's generally OK. A new boyfriend or girlfriend will usually qualify as a "competent adult" unless they have an extensive criminal background for violent offenses, drug offenses, have a protective order or a history of protective orders, or are a registered sex offender. Most low-level misdemeanors will not be enough to exclude new boyfriend or new girlfriend from handling pickups and drop offs. Contact an attorney in your area to determine the specifics in your case and what rises to an worrying level for the Judges that may hear your case. Yes, this also means new partners can handle the exchanges alone without the actual parent present.
A word of caution: it is best not to let new wife, girlfriend, husband, boyfriend handle communications on your behalf about the children. Concerns about the children's school or extracurriculars should be handled by the parents themselves. Details about the visitation schedule and any modification requests should be handled by the parents themselves. Allowing a new partner to handle these topics with the other parent is asking for trouble. Judges prefer the parents to handle the communications about the children, not new partners. If you are allowing your partner to handle these communications because you "just can't have a reasonable conversation" with the other parent, then that's going to be a mark against you in the future. Finding a way to communicate for the good of the children is in everyone's best interests. There are many short-term counseling seminars, online communication tools, and other options to explore before you throw in the towel. Try these options and you will be in a better place for your next court hearing or maybe avoid one altogether. If the other parent is being truly unreasonable and demanding in all your attempts to have a reasonable discussion about the children then reach out to an attorney in your area about options. There are change that can be made if you are experiencing truly poor behavior from the other parent.