Does the room he is in have light and good air flow? Does he get a quality, timothy-hay based pellet food, unlimited hay, and daily veggies, along with attention so he isn't lonely? If so, he should be fine for a few more months. Just try and give him some floortime every day if you can. If he is nervous for a while though, that's okay! Start floortime once he's used to you being around him so floortime doesn't become scarytime*. When he is ready for floortime though, just corner off an area of a large room or use a small room with a door and let him out so he can run around. It helps to have a few places where he can hide too, both in and out of his cage so he has a few safe places to go. I'm so glad you could take this guy though--he's very cute and you sound like a very loving owner, he's lucky to have you!
Also, I'm not sure about your parents, but my own mom is the most stubborn individual I know (sorry mom, I love her but she's stubborn as can be) and she was VERY much against me using a c and c cage. She only said this after I told her about it for months and then built the thing, which was annoying, but I managed to convince her by proving how happy Bob was in his big cage. But if they do not want you to make your own cage, there IS a cheap-ish premade cage that could work for you called a midwest cage. It's a bit flimsier than a c and c and has a leak-proof canvas instead of a plastic tray, but it's the appropriate size for one pig and two can be connected to make a bigger one. How that is done, I do not know, but I do know it can be. So if they offer to buy you a premade cage instead of a DIY one, that is a very good option. If they are up to you getting him a bigger cage but not either of those, please tell us! There are so many fantastic options for cages out there...most are DIY mind you, but they're viable, like a melamine or plywood cage, or that one plywood and door cage someone made not too long ago. But my point is have options, and show them examples of cages people have built to show them what you mean. Sometimes seeing is believing, you know?
And if they are up for you getting a c and c, then FANTASTIC! Just let them know being a guy, he and his guy friend will need a cage equal to a 2x5 or a 3x4 c and c to be comfortable (boars need more space than sows, you see). But gosh, the change in happiness and health is astounding. He'll be a lot happier with more space, that's for sure! And I'll take a little look around to see if I can find somewhere to get c and c supplies in the UK, but I make no guarantee--I live in the US and wouldn't know where to find anything, but I can research.