Question

One word to mediate between two parties in a dispute meaning

Answer 1

Mediation is a dynamic, structured, interactive process where a neutral third party assists disputing parties in resolving conflict through the use of specialized communication and negotiation techniques. All participants in mediation are encouraged to actively participate in the process. Mediation is a "party-centered" process in that it is focused primarily upon the needs, rights, and interests of the parties. The mediator uses a wide variety of techniques to guide the process in a constructive direction and to help the parties find their optimal solution. A mediator is facilitative in that she/he manages the interaction between parties and facilitates open communication. 

Answer 2

One word substitute is Intercede. Interfere: intervene in a situation without invitation or necessity. Interact: act in such a way as to have an effect on each other. Interrupt: stop the continuous progress of (an activity or process). Intercede: intervene on behalf of another.