With the latest versions of Android, they made sideloading apps way more difficult so I wouldn't be surprised if your parents weren't able to find how to it. Also, if you really want to block it you can use parental controls.

We effectively have a duopoly. It's impossible for a third operating system to gain any traction and until that changes we're stuck with these two options.

A lot of things nowadays rely on apps so not having a phone with either Android or iOS means it's increasingly hard to do stuff. For example, with 2FA regulation in the EU for banks, they effectively made their mobile apps indispensable for being able to make some payments. I shouldn't have to use either iOS or Android to bank but here we are. It's even worse when you can't even use a custom ROM because the apps refuse to work on one.

You're right the way that they should force Apple and all the other phone manufacturers to open their bootloader and provide a common interface so that we can finally have some real freedom and competition on the mobile OS market. We should not settle for anything else than the freedom we already have on PCs.