I have started writing lyrics to a tune which undoubtedly has copyright (one of the 'Sound of Music' songs).
The lyrics are a brexit parody.

I would be very surprised if I'm allowed to publish it as it uses the old tune and much of the rhyme scheme - though totally different words.

Would it be permissible if I made the music myself but to the same tune (or very close)?

I wouldn't want to steal intellectual property but I'm unclear what the legal situation is. I say this because, many years ago, I bought a tape with new lyrics to popular songs. They were songs about an industrial dispute in the printing industry - songs like 'Build a Bonfire' (putting Rupert Murdoch on the top), and lyrics like 'If you think Rupert Murdoch is an old shitbag then smile, smile, smile. If you think the Sun is just a filthy rag, then smile, smile smile, smile smile etc (that was to the tune of that one about 'pack up your troubles in your old kit bag ...'

Then there was 'Waltzing Matilda' when they sang '...and they sent me away to the dole...' (rather than sent me away to the war) - the other lyrics were changed too. There was a whole tapeful of them.

So it has been done, but maybe no-one owned the copyright. Does anyone know how I could clarify this situation?