Btw, what do they rent it for? I didn't know that was a thing!
Occasionally it's a lease of an expensive competition or breeding animal that the people couldn't afford to buy.
Usually it's a loan, ie not renting as such and no payment goes to the owner, but the person taking on the horse pays for all the expenses related to it's care. It's a way of retaining ownership of a horse you can't afford to/don't want to keep any more. So you can keep an eye on its welfare and take back or PTS if there's any issues, you lose this control if you sell. You may also know in advance you want the horse back at some point eg after pregnancy, when the busy phase in your career is over, when you return from travelling or university etc.
The person taking the horse on effectively gets a horse for free.
The downside is they don't own it and could get emotionally attached or put a lot of effort into training it, then the owner decides they want it back. They may have to comply with terms set by the owner regarding use of the horse (activities it can participate in) and methods of keeping it. Usually these specifications concern the horses welfare and best interests, sometimes they concern an overly fussy owner and are a nuisance.
The upside is they don't own it, so if the horse becomes unable to work through age, injury or infirmity, or their life circumstances change meaning they can't afford/look after/ride a horse any more, they can simply hand it back to the owner if they wish. They also don't incur the costs to PTS at end of life, which can be considerable.
OP IANAL but I suspect you won't get anywhere with this.
I understand your parents bought a horse "for you" but realistically if they never officially gifted it to you by signing a receipt transferring the horse to you, or putting the passport into your name etc, then it looks as though legally they intended to retain ownership themselves. You've got nothing to prove the horse is yours. Did you ever pay for all the expenses of keeping it yourself and have full and final decisions on its care? If you did and can prove it perhaps you might get somewhere arguing that it's your horse. Unfortunately I don't think you'll ever see the horse again.
Your parents have fallen for a common scam. Horses aren't sold on payment plans because generally people who ask for that don't have the money to buy a horse, don't have much spare money at all (for all the non-horsey people reading, it's a total myth that horse people are rich, most are totally skint after basic horse related costs and basic human life related costs. Plenty earn only minimum wage, don't really have a career etc) and have no desire to part with what they do have. They secure agreement for a payment plan, pay some of the money, then cease payment, claiming they now own the horse and have paid for it. As you say, the shoddy contracts people draw up aren't worth the paper they're written on and possession is 9/10 of the law, as the saying goes. They have most likely already moved the horse to a new location.
Nobody really ever enforced horse passport rules anyway, the government only brought them in in the first place due to an EU ruling about food chain animals (which in many countries horses are). Post Brexit I image the government cares about horse passports even less than it did before. The person who has the horse is unlikely to care about it at all, it's not going to stop them having the horse treated by a vet, PTS, sold etc and if they've no money I don't suppose they'll be into competing at a high level where the governing bodies of the sport or the people running the competition might want to see the passport.