According to Wikipedia, the location of Vaidyanath Jyotirlinga is given at two places: Parli in Maharashtra and Deoghar in Jharkhand. Which of those is the correct location of the Jyotirlinga?
The location of the Vaidyanath Jyothirlinga is said to be at 3 places, not two. Swift's beautiful answer enumerates them as:
1) Vaidyanath Jyotirlinga - Deogarh: -Baidyanath Dham , Deogarh district of Jharkhand.
2) Vaidyanath Jyotirlinga - Parli Village , Parli Vaidyanath temple , Beed District , Maharastra State
3) Vaidyanath Jyotirlinga - Kangra:- Baijnath , Himachal Pradesh State.
What exactly is the dispute?
To know the location of the jyothirlinga, we can utilize three resources. Firstly from a verse which Sri Shankaracharya has composed (again taken from Swift's answer):
पूर्वोत्तरे प्रज्वलिक निधाने । सदा वसंतम गिरीजा संमेतं ।। सुराससुरधित
पादपद्मम । श्री वैद्यनाथ ताममहं नमामि ।।
Poorvothare prajwalika nidhane sada vasantham girija sametham
surasuraradhitha padapadmam srivaidyanatham thamaham namami
Here, the first word "Poorvothare" means North East. Secondly, from the Shiva Purana. (again quoting from Swift's answer):
According to Shiva MahaPuran KotiRudra Samhita 1:21:24 (शिव महापुराण कोटि रूद्र संहिता) the location of Baidyanath (वैद्यनाथ) is identified as"chidabhoomi" (चिदा भूमि) - which is possibly the ancient name of Deoghar (देव घर).
And finally from the Dwadasha Jyothirlinga stotra:
Paraly Vaidyanāthaṃ cha Ḍākinyāṃ Bhīmaśhaṅkaram
These give us a brief idea about where the Jyothirlinga is located. To recap, Vaidyanath is present in the North East in a chidabhoomi at a place called Paraly. Now iterating through the 3 places:
Vaidyanath at Deogarh - The ancient name of the place is "chidabhoomi" and is present roughly to the North East. Hence there are two connections.
Vaidyanath at Parli - The name of the place is itself Parli, and it was a place where bodies were cremated. Hence there are two connections here as well.
Vaidyanath at Kangra - The place is located next to a place where they used to cremate bodies. Hence utilizing the "chidabhoomi" connection, it is claimed that the place is the original jyothirlinga.
Which is the correct location of the Jyotirlinga?
We can never say which one is the correct place. All three have their own background stories, and are equally regarded as the Jyothirlinga. They all are very ancient temples, because of which there is no use of carbon dating or any other scientific technique to discover the oldest amongst the three.
All the two temples are ancient and having old history. But when It comes to jyotirlinga, Baidyanth dham Deoghar Jharkhand is among them. It is present in shiva Purana that the place deoghar is "Chitaa bhumi",It means the last rituals of Godess Sati was completed there.So it is considered as the favourate place of Lord Shiva. Baidyanth Dham Deoghar Jharkhand is among 12 jyotirlinga.