B737-800 Captain reported the parking brake indicating light illuminated in flight. Flight continued to destination and landed safely.

Narrative

About 45 after departure; level at FL370 the parking brake light illuminated. Conferred with Dispatch and Maintenance via crew phone. Determined light received signal and current from a brake SOV (Shut Off Valve) as well as a parking brake set" relay. This confirmed to me the possibility of engaged parking brake at landing. No checklist in QRH. Used "tire failure" information and pilot reasoning for discussion. Crew discussed tire failure; brake fire; and complete loss of aircraft control. No immediate safety of flight existed. About 3 hours remaining in flight. Weather in ZZZ suitable. Developed contingency of ZZZ1 if weather degraded (temp and due point spread 2 degrees). Relayed; to dispatch; intention to [request priority handling] with ZZZ center in a few hours. Continued to destination. 1:20 prior to landing; coordinated necessary preparation time with #1 FA (Flight Attendant). Performed all call and gave briefing. Roughly XA:25. Elected to defer PAX (Passenger) PA to XB:00; 45 prior to landing. Made PA and FAs completed pre evacuation duties; moving passengers away from wings (brake fire location) towards door exits. Upon initial contact with ZZZ center; [requested priority handling]; requested ARFF (Airport Rescue and Firefighting) and runway XXC with long final and early descent to 5000'. Transferred aircraft to CA for landing. Depressurized cabin per procedure at 5000' and FO made "Brace" at 250' AGL. On landing no abnormalities were observed. ARFF confirmed brake temps and tire conditions normal. Light was illuminated on touchdown and rollout. After setting of parking brake for passenger PA; light extinguished on next parking brake release. Taxied to gate uneventful. The entire cabin crew deserves high marks for professionalism and being a useful and helpful part of the process. They used their own creative reasoning (given the precise mechanical situation) but conferred with the Captain to complete the communication circle. I know it was unusual and stressful at times. It was burden-lifting to be able to give the briefing and have confidence the passengers were prepared for the grave possibilities at landing. The cabin department should know of a job well done for training preparedness and leadership shown by my crew. Cause: Mechanical failure. Risks were mitigated with quick; frank; accurate and respectful communication via ACARS and Crew Phone with Dispatch and Maintenance. Commendable professionalism from cabin crew and first officer as well!"