The answer turns on who is present when these comments are being made. If the children are not present, the court cannot make an order restraining a parent from making comments regarding the new significant other as that would constitute an undue restraint on speech. However, if these comments are being made in the children's presence, the court could make an order to "not making disparaging remarks regarding the new significant other in the presence of the minor children or permit other 3rd parties from making such comments." The reason being that the making of such comments may undermine the relationship between the child and the step parent/significant other. However, the parent will need to prove that such behavior is occurring without necessarily involving the children to provide such proof. For example, if a parent posts a derogatory comment about the step-parent on Facebook for which the child has access to this account.
The other parent bad mouths my new significant other/ step-parent Is there anything I can do to stop or limit this behavior?