I know aaaall about this sort of thing so if you want to PM me with direct questions, or ask me on this thread, feel free. I came out in 2017 just before I started my A Levels, and I've now changed my name on my driving license (not passport yet, as to change my gender on that I have to provide proof from my GP that I have lived as male for over two years
) and have applied to university as male. Granted, I said on my application that I was transgender because they asked, but apart from that I don't plan on telling anyone who doesn't need to know. As
@The Empire Odyssey said, universities do love their legalities and regulations. However, there is a way for you to change your name without your parents knowing, assuming that you're 16 or over. Head over to
https://www.gov.uk/change-name-deed-poll/make-an-adult-deed-poll and there's a template on there to write your own deed poll. You will need two witnesses though (can be absolutely anyone so long as they're over 18 and not a family member - could be trusted teachers, friends, friends' parents, etc.). I know it seems a bit dodgy that you're allowed to write out a legal document yourself, but that's exactly what I did and it was accepted by the DVLA to change my license and by the HM Passport Office.
Basically, coming out to your parents isn't necessary to change your name. I applied on UCAS before I had actually changed my name on anything, because they won't need proof of identity until you apply for student housing. I'd get your name changed ASAP, and if you haven't applied to UCAS this year, when you do use your new name. If you already have applied, I'd wait until you accept your firm and insurance and then email the universities individually to let them know of your name change. Feel free to ask me any more questions on here or on PM!