Adoptions and where parents are unknown are usually very hard. It seems likely his adoptive parents were Stoughton and Margaret Barber as indicated in the 1900 census.
The real question is was Barber his birth surname or his adopted surname? Also did the Barbers legally and formally adopt him or simply raised him as their own?
I suggest looking at his marriage application which should be on record at the Luzerne County Orphans Court Office. The application is different from the marriage record as such. The latter just records the marriage, the former asked questions about parents, etc. The Orphans Court may also have any guardian record for William assuming he inherited anything as a child. If his parents died while he was young this is possible. If he inherited anything as an adult he might appear in an estate file or deed selling property. Check with the Register of Wills and Recorder of Deeds Offices. I would at least look at the estate papers for his adoptive parents and other Barbers from back when. Adoption records in Pennsylvania are unfortunately literally sealed forever and can only be opened by a court order no matter how long ago the adoption took place. Be careful about mentioning adoption at the courthouse. In some Pennsylvania courthouses the personnel are very secretive about any dealing with adoption even ones from over a century ago. The Courthouse was flooded in Hurricane Agnes in 1972 which did a lot of damage to the records there so don't be surprised if there are gaps in the records. However, some were microfilmed by the LDS Church before the flood.
If he was baptized as a child his natural parents may be listed in the record. The problem is knowing which church they belonged to. Presumably he was baptized at a church in or near Plymouth.
When you say his obituary says his parents are unknown do you mean it literally says they are unknown or do you mean they are not listed? Back then it wasn't entirely common to not mention the parents' names in obits. Presumably it also doesn't mention siblings. His death certificate may still have his parents' names. It might list them as unknown, have his adoptive parents' names or if you're real lucky his natural parents (in other words don't assume it doesn't).
Check with the Luzerne County Historical Society and the Northeast Pennsylvania Genealogical Society. Plymouth does have its own historical Society. I don't know if they can help you with this or not.