"There have been multiple times throughout my career when I've encountered two parties who disagreed with each other. In fact, it happens all the time! Many times resolution can be found after I encourage respectful discussion and compromise. Other times, I have to bring in a stakeholder to hear both sides and make the final determination.
One scenario in which I would escalate the issue is if there is a resource conflict. For example, if employees are being pulled off project work to handle operational work instead. Naturally, this kind of issue can threaten the project's timeline so, in this case, I would reach out to a stakeholder to explain the urgency and importance of the project and to agree on a workable solution, which might be gaining a replacement or accepting a change in the project's timeline. Whatever the outcome, it's also up to me to communicate these changes and/or solutions to everyone."