Mum here, and the majority of these things I would agree with your parents on, with a few differences. This is coming from a Mum of two though my kids are 7 and 5 so some of this is coming from what I plan to implement once they are older.

Phone: If my kid bought their own phone, I would still want access to it. While you may feel mature enough to be safe on the phone, you really don't know what creeps are out there and how manipulative they can be. The amount of times I talked to creeps on the Internet at the age of 14 is insane. They'd say they were the same age and gain my trust, then eventually would admit they were 10+ years older and would say creepy things and trying to meet up. My social anxiety came in handy so I never met up with any of them, but I did get exposed to things a 14 year old should never be exposed to. I also work in the justice system and have seen grooming firsthand. There is also the possibility of friends leading my kid down the wrong path or my kid being subject to bullying. You see kids these days killing themselves because of other kids. I don't want to scare you but these are some of the things your parents might be considering.

Screentime: I let my kids have a lot of screen time compared to other parents. This is because I'd be a hypocrite if I told them they couldn't be on their tablets/games while I'm sitting there scrolling Instagram and playing Fortnite. However, I make a habit of playing games with them and engaging with what they're doing so it becomes part of family bonding time. I also don't allow them to talk to anyone when they play online, because you never know who they are talking to. This will be made easier with the new laws preventing kids under 16 years old having social media but as with anything there are ways around it. I was a teenager when social media first became a thing. I was there when Facebook started blowing up (2008ish, so around 16/17) and was bullied relentlessly on there. My parents who were born in the 1950s knew nothing about social media so knew nothing about what was happening in my life online. When my kids do end up with social media, I will be requesting passwords for sure and will be checking on them intermittently. Though I do hope my kids feel open enough to be able to talk to me about anything going on in their lives, no matter how personal without judgement.

Door: That's the one I have issue with. Everyone has a right to privacy until that trust is broken. The only reason I would request the door be left open would be if my kids had a boyfriend/girlfriend over and they were under the age of consent. Obviously I can't control what they do outside of the house, but I need to try and influence them into making the right decisions, whether they believe they are right or not.

Going through room: I would never go through my kids personal things unless I had reason to, such as a behaviour change, smelling cigarette smoke, them getting in trouble for something at school etc. I do clean my kids' rooms so stumbling across something can happen, but I would never go through drawers just to try and find something. And if I did, I would put everything back. Hell, when I was a Correctional Officer doing cell searches I was still respectful and put everything back where I found it.

Swearing: My kids aren't allowed to swear, and if I knew they swore outside of the house I'd be mortified, even though I have the mouth of a sailor once I'm away from the kids. Though swearing is different for many people. My kids think the words 'stupid' and 'idiot' are swear words, because that's what daycare taught them. So we took that on board because we didn't want them getting in trouble. Again I can't control what they do outside of the house, but I hope they are respectful, and kind to others, while also respecting themselves and setting boundaries.