My mother wants a mini too, and I keep telling her exactly what your parents are telling you.
What pray tell does she think she's going to do with it? Then again, my mother also wants a donkey, a pigmy goat, an alpaca, and a teddy bear Pomeranian. The list goes on. And I keep telling her no.
I suppose you could claim to have epilepsy and say that the mini detects seizures. That way, you could take him everywhere. There are dogs who do that, you know. Why not a horse? Or, you could tell your folks that minis are much cheaper than normal horses, and you would be just as happy with a mini, so... yeah.