Parent starting to get on my case for being lonely

Your parents are right.
You are missing out on the joint most pleasurable thing in the world.
You are missing out on enhancements to most of the other joint most pleasurable things in the world.
Your parents love you and want the best for you.
Instead of shutting down when they bring this subject up, you should embrace it.
Finding love and keeping it is important. They're fascinating subjects. They're the sort of things you should be having interesting / fun / useful / productive conversations with your parents about.

For the longest time my parents didn't really comment on things like the fact that at 24 I've never had a girlfriend, or that I don't hang out with any friends, etc. however that has started to creep into conversation recently. Even their birthday wishes essentially boiled down to "hope you find some friends". Other times it's directly asking me if even when we move house I'll continue being as lonely as I am now. And every time it did I was so stunned I didn't really know how to respond. It's not in an overall critical or harsh way, but one that still irks me, and again I just have no idea how to react or what to say to them when they start asking about it. Mostly I just double down and don't play along. Anyone got any similar experiences?

Don't care if they're "fascinating subjects" or whatever. It's not something I can do any about so talking about it is pointless.

There's nothing wrong with being lonely as long as you are independent and happy. Maybe gently letting them know that you are happy where you are in your life and reassure them you are happy with your solitude.
24 is still fairly young and you will explore friendship groups etc when you are ready, maybe letting them know that will help them refrain from pushing the topic onto you.

Is the issue here you want to make friends etc, but have been unable to/feel unable to so far? Or that you genuinely just have no interest in making friends and connecting with others and are ambivalent about other people's company?
It sounds like the former given you are also bothered by your parents bringing it up, but it's not wholly clear.
I ask as the latter might be something to talk to someone about that. If the former then I would suggest avoiding "doomerism" about it and just try and make connections where you are able and the opportunity presents itself - sometimes it's just a case of attrition.

I guess I'm in the camp of "it would probably be nice to have friends or a girlfriend or in general talk to people or something, but there's nothing I can do about it so I may as well not even think or worry about it or let it bother me".

Why do you say there's nothing you can do about it? What is it that prevents you from doing that?

I get it. I have no friends either but I find it comfortable and peaceful. My mum always says 'why don't you have friends?' or 'You need friends' . My personal favourite is 'You don't have friends because there is something wrong with you'. I think your parents are just worried.

I don't really have any real opportunities to meet anyone or any opportunities to make those opportunities if that makes sense.

Well usually to meet people after leaving school you need to find some kind of social activity to join up with and go consistently to it - really any kind of social group/club or hobby group works. Could be anything - bridge, community theatre, bingo, LARPing, tabletop gaming nights, sports, creative arts groups/classes, etc. Most towns will have some stuff in that vein being run by some person or group so have a look around. If you're at uni then you get a slightly more convenient platform to access these via university societies and such, but even if you're not at uni and are a working professional those things do exist and people do them!
And pretty much by just showing up consistently you start to get to know people, and from there develop friendships and potentially romantic relationships even. Of course it does require the engagement from you to seek out those opportunities and make the time to go to them and be disciplined about making time for it to go consistently (whatever the frequency is - daily, weekly, monthly, etc). The consistency is what makes the difference.
Post-school though that's the only way anyone makes friends really, you're no longer being locked in a classroom prison for 6 hours a day with other people you get to know. Arguably there is your workplace but it can be harder to get to know people at work and/or the workplace dynamics can make things awkward (whereas at something you're doing purely independently in your free time, you don't have that).
I assure you there are some opportunities to go and do something like that - it doesn't even have to be something you're deeply passionate about, it can even just be something you think would be vaguely interesting and through which you'll meet other people.

Even if I did go anywhere I still wouldn't know how to actually make connections with people. And work-wise that's not an option either as I work from home.

You don't need to really do much other than consistently turn up to something and interact with other people. There's not a magic formula for it, that's all you (can) do and then sometimes someone you interact with you'll develop a friendship with.

And again I don't know how to begin interact with anyone.

You're interacting with someone right now
So...just like that!

Let's not even pretend this is anything like striking up conversations with people in the real world.

I think I can only see so many variations of "I wouldn't even know where to start", and people giving cracking advice on the myriad ways to start, before I accept that someone can't be bothered. Which is not in of itself an issue, unless they just want to bring it up ad nauseum.
Basically it's either issue that you want to work on, or it isn't.