You will not have any trouble with border crossings of this nature as a UK citizen. For example, border control in Brazil will have no idea where your parents are from and won’t care. They will see your UK passport, and let you in for 90 days visa free, just like any other UK citizen.
If you have no plans to visit Turkey or North Cyprus, then you will have nothing to worry about. If you do, then the details of what happens depends on whether they consider you a citizen of their country, and what their rules are for mandatory service.
Your UK passport offers you a great many places to visit with no visa required. For the rest of the world, visitor visas are usually pretty easy to get.
Credit:stackoverflow.com‘