Not if you don't want it to. I'm about to start a PhD at Loughborough but I teach at a different uni. We have very strict policies about protecting privacy around this kind of thing, and generally:
(a) during term time stuff will go to your term time address
(b) we prefer email on our internal systems (you'll get an @lboro.ac.uk address soon)
Just tell the university that you want to be paperless, but they're more than used to supporting students who don't want "outing" for various reasons (we had a girl who got pregnant and went part-time at university with a term off whilst baby was very little... she was just as nervous as you are and so the university made sure that we communicated in her preferred way).
Parents are often loving but confused by this kind of thing: think about buddying up with someone supportive from the student's union/LGBTQ+ community on campus and getting a steer on what comes next.
Remember, you're an adult now, and good for you for making this brave step.