Air Carrier First Officer reported a fume event during climb. After being notified by flight attendants that the odor was also detected in the cabin the flight crew requested priority handling to departure airport where support personnel met the flight.

Narrative

Our flight was due to depart at XA:00; ZZZ-ZZZ1; and was filed to fly the entire route within and just east of an advancing frontal system that stretched the entire length of the east coast. Conditions for our departure were IFR with light precipitation. We departed XXL with a tailwind; Takeoff Go Around; APU ON; ANTI-ICE ON; CONFIG 2; and immediately found ourselves in widespread light to moderate precipitation; light to moderate chop; and a need to begin deviations for weather as we proceeded on course. We cleaned the aircraft up; including turning off the APU bleed and APU as normal. While climbing through 10000 ft.; I noticed an unusual odor that was gradually increasing in strength. I asked my Captain if he detected it also; and he said yes. Before reaching FL180; the #1 Flight Attendant called the cockpit and stated that he smelled an unusual 'oily; burnt hydraulic fluid' smell. I concurred and thought that was an apt description. My Captain asked him to try to determine if it seemed to be getting stronger; dissipating; or staying the same; and if it were possible to see where it may be coming from. After reaching FL250; or just a few minutes later; several of the Flight Attendants called back at the same time and informed my Captain that they found the smell to be at least as strong; if not more so than previously reported. I explained to him that I had a good understanding and awareness of Smoke/Odor/Fumes events; and it was my opinion that we were experiencing one. He then asked me for my input about how we should proceed. I said that I felt it would be wise to [put on] our oxygen masks; land without delay at the nearest suitable airport and run the Smoke/Odor/Fumes and Overweight Landing checklists on the way to that airport. He agreed and informed the Cabin Crew of our intentions while I took the radios in addition to flying duties and [requested priority handling] an immediate return to ZZZ. While making our return to ZZZ; the widespread precipitation conditions we were in were such that our weather radar displayed mostly green returns; with occasional yellow interspersed within which would normally indicate light rain; but the very loud rain noise made it obvious that we were in at least very heavy precipitation. This phenomenon was making the radar of not very much use other than to confirm that there was rain in front of us. Constant light to moderate chop persisted through the entire event; also. Another trait of the Airbus 320 family is that the VHF radios produce a great deal of loud static when flying through precipitation for extended periods; and particularly in the vicinity of charged clouds; which in this case made ATC difficult if not impossible to understand at times; and added to the now very loud environment of the cockpit. Despite these challenges; we were able to comply with ATC's directions and make a nearly direct path back to the ILS for Runway XXL (which by now was no longer experiencing a tailwind for landing). My Captain recognized that we would be back on the ground soon enough that he had to prioritize which QRH checklist to execute; the Smoke/Odor/Fumes checklist; or the Overweight Landing Checklist. We chose to perform the Overweight Landing checklist first and found that when we turned the APU and APU BLEED on as the procedure directs; the odor began to abate. We were satisfied that this configuration of the air conditioning system was now the optimal one and decided to continue to land without further reference to the Smoke/Odor/Fumes checklist; as time before landing was now quite short. I made a relatively soft landing; approximately 12000 lbs. overweight without any other irregularities; and we taxied the aircraft to the gate normally. There were no excessive brake temperatures. By this time; we could scarcely detect the odor anymore; and I felt comfortable to have my oxygen mask off for the duration of the taxi. It may be that we had begun to acclimate to the smell; because when we parked at the gate and opened the main cabin door; the Mechanics who immediately boarded the airplane stated that they noticed the telltale smell right away. Appropriate Maintenance logbook entries were made; a Smoke/Odor/Fumes form was completed; and much discussion with the Mechanics took place about what we experienced.As of yet; I haven't heard any confirmation of what caused the event in this airplane. I could only speculate that oil or hydraulic fluid found its way into the bleed system of one or both engines and was transmitted as a fine vapor through the packs and into the cabin. It seems that the APU may not be a likely culprit in this case as the smell wasn't noticed by us during taxi-out; take-off; or during our overweight landing. The APU BLEED was supplying air to the packs in all three of these instances.I think that this is an area where it would be best to solicit suggestions from Maintenance Personnel and the manufacturer. I have what I would judge as a heightened awareness of these events (especially now); but not a complete understanding of the specific Mechanics of these events as it pertains to the engine/APU bearings; oil systems; etc. The industry; company; and pilot unions seems to have taken significant steps in recent years to address these events and raise awareness; but my suggestion would be to pursue these steps further. I have flown with many pilots who have limited knowledge of the prevalence and risks associated with Smoke/Odor/Fume events; which is surprising considering that (based on my limited perspective and data) these events seem to rank high among drivers of QRH usage and diversions; etc. I think including information about them in distance learning modules would be a good idea.