Parmesan is the English and American translation of the Italian word Parmigiano-Reggiano.This isn't the first time I've seen something like 'the English and American word for X' or similar, but it is jarring to me.
The thing that makes me have to stop and re-read it is of course that I'm interpreting English as the language, while the writer seems to be treating it as the country, and simultaneously conflating England and Britain/the UK. I know that to people in other countries, it doesn't seem very important that England and Britain are not the same thing (in Britain, we do a similar thing with Holland/the Netherlands) and furthermore, there is no adjective relating to the UK (UK-ish?), which would really be the more correct name anyway (to include Northern Ireland). But to me, England and the UK are not at all the same thing, and it is odd to say that something is English when it's clearly also British. This is a pragmatic implicature: you should give as much information as you are able, and here you're withholding some information (namely that the term is also used in other countries). And then what about Canada, Australia, Ireland and so on? Surely it's better to refer to English as a language here than to be unnecessarily specific about countries.
But I think there's more to it. Look at how it says 'English and American translation'. Most Americans are not idiots, contrary to the national stereotype that we like to believe in, and they know that they speak English rather than American. But a mixture of strong patriotism and some genuine confusion (actual people I know have been asked what language they speak when visiting America) means that it's sometimes tempting to write 'and American' to avoid people thinking you're just talking about England. I think many people probably do half-think of American as a language, even if they know that it's English really. And, of course, we might classify it as a separate language, if we so chose: language boundaries are not only determined by mutual intelligibility, as any linguistics student knows.