That's a very fair question. The answer is that there have been several court cases in which this very dispute has been at issue – besides George Harrison's 19th-century case, the main recent test case is that of Tim Saunders (around 1980ish? – I've got the date somewhere or other), who argued he didn't have to pay the poll tax because the Stannary Parliament had vetoed it in Cornwall. His case was thrown out, so the dispute remains unsettled. Besides the revived Stannators themselves, I'd actually suggest the people of Cornwall as a whole are party to the debate – and I don't just say this as a Cornish nationalist, Britain's a democracy, so its constitutional questions are a concern of the people.
There's also a sense in which it doesn't matter whether Cornwall is an ordinary county or not, and that is the question of whether we should think of Cornwall as part of England in any kind of non-legal sense. Think about the status of Merionethshire in the 1880s. Legally it was just an ordinary county of England-and-Wales; yet it clearly belonged to a specific 'country', if you like, namely Wales. Conceivably, then, Cornwall could be (legally) an ordinary county of England-and-Cornwall, but a 'country' of its own in a different way.
I know what you mean, in a sense – it doesn't seem like much of a dispute if Cornwall County Council isn't taking part. But look at all the controversy surrounding this very article (and the first paragraph of Cornwall, from which this article springs) – if it isn't a dispute, what do we call it?