In light of the recent Brexit vote, I was curious as to whether parish councils could also call for something similar, with governing borough councils if they were seen to be negligent of said parish. For example could a parish council legally either trigger some legislation to move to another borough (or even become an independent district), or would this require a referendum of sorts, if at all possible.
With the caveat that I do not believe any parish council has declared independence so the legal situation hasn't been tested, I believe the answer is as follows:
There is no legal basis for which a parish, borough, county or district can claim independence. That doesn't, however, mean that the law couldn't be changed to allow it.
On a larger scale, it is fundamentally no different to the situation with Scotland and the call for independence there. Currently, the law does not allow for Scotland to become independent, unilaterally, even though it has its own government. For it to become a properly independent nation, the Westminster parliament would have to enact the relevant legislation.
Brexit is somewhat different because there is an explicit mechanism for a nation to leave the EU: the now famous Article 50. Even there, the wording is commonly assumed to require an act of Parliament to invoke although this is by no means a universal interpretation.
One last point of clarification. In neither the Scottish nor EU independence cases was a referendum sufficient to actually invoke independence. In both cases it is legally, though probably not politically, possible for the government of the day to set the referendum result aside. So, unfortunately, the parish council independence referendum wouldn't be sufficient to declare the Independent Republic of Upper Hazelfordmere. Mores the pity.