Hi welcome to PC.

It sounds as if your parents see a stigma with mental I'll health, even though at least 1 in 4 people experience it.
My parents used to have similar though less extreme attitudes and it took years of life experience for them to begin to understand and longer before I could discuss any aspect of my depression with them.

You are an adult so you can make a gp appointment for yourself and noone needs to know.
Maybe your boyfriend will go with you for moral support (he can stay in the waiting room or go in with you, whatever you want).

It could be worth while asking at reception or looking online to see which doctors at your surgery have an interest in mental health to have a better chance of a positive experience, but many doctors are sympathetic.

There is no shame in living with depression! Many of our great people have done (Churchill and Stephen Fry to name just two).

Having depression on your health records will cause no problems. It may help doctors care for any physical health problems better if they understand you're vulnerable. Nobody outside your gp or mental health practitioner need ever know unless you tell them.
There are options for help from gp and IAPT the primary care access to talking treatment.

I wasn't diagnosed till I was 19. I had left home by then, but getting help really made a difference.

Lecture over! Whatever you decide I wish you all the best and good luck with your efforts.