B737 flight crew reported exceedance of cabin pressure differential limit in cruise. Flight crew diverted and landed uneventfully.

Narrative

Whenever I reach cruise flight; I always scan of all the aircraft systems. When doing this I noticed that our pressure differential was into the amber band. I asked the Captain if this was normal as I have never seen this before on any aircraft type that I have flown. We then referenced the QRH only to find that there was no procedure for this issue. We referenced the limitation section of the Flight Manual; the plane had exceeded the pressure differential limit. We were around 100 or so miles from entering class II airspace and opted to do a hold to stay within the vicinity of ZZZ in case we needed to divert as well as stay in radio and radar range. At this point we spilt the flight deck; I remained pilot flying and took over the radios while the Captain contacted ops. Between talking with ATC when the worked load permitted; I would listen to the conversation between the Captain and operations. I was shocked to hear that they wanted us to continue to ZZZ1. I did not chime in; but I was completely floored that this was what we were told to do. The Captain repeatedly stated that you are asking us to fly with a known problem one that could lead to a structural issue which could cause a rapid decompression. While this is happening; we continued to request lower altitudes due the continued rise in differential pressure. After the Captain talked with Ops he concluded the radio patch and fully updated me on what was said. I only heard small parts on the conversation as I was only able to listen for a very short amount of time due to my workload as PF and talking to ATC. The Captain then said that they told us that we can continue to ZZZ1 with this pressurization problem. I asked the Captain why are you going back and forth with them you know what the right answer is. I feel he only second guessed his right decision for just a few moments due to the fact (we were told it was OK). There seemed to be a tone of pressure to complete this flight. We exited the hold; another patch was made; and we were told to climb back up to our cruise altitude because this would allow for enough fuel to complete the flight to ZZZ1. I cannot understand why the captain is being told that it is OK to fly the plane with a known problem. The flight routing; the issue with the pressurization; the type of navigation all points to this is not a good idea. Continuing the flight to ZZZ1 only would have open the door for a multitude of problems that could have ended horribly. Once on the ground in ZZZ2; Maintenance told us there was an issue with the outflow valve; pressure controller computer and some grounding wires. They also stated that we would not have been good to continue to ZZZ1. In later research this same aircraft two days later had another pressurization problem forcing the crew to utilize the manual function of the pressurization panel. I am writing this not for myself and the Captain but for the tech who stated that it was OK to continue to ZZZ1. I feel the advice given was not well thought out. Climbing to a higher altitude with an over pressurized plane would only make the situation worse. If we have limitations in our Flight Manual we should adhere to them and not be told that its OK to continue when there is an exceedance. Safety and precaution should have been the focus.