I don't think that would keep a rabbit safe from predators, unless you put it inside something else (like inside a garage, or inside a chicken run.) A raccoon or cat or something could reach right between those bars to get it.
And I agree, that's too small to keep a rabbit for more than a few days.
But I do see your parents' point too, that you don't want to spend the time & money to create a good cage and then have the rabbit disappear.
Maybe you could do the research for where to buy or how to make a suitable cage, get their agreement to make it happen as soon as the rabbit is caught, then trap the rabbit with the cage you have. That way the rabbit won't have to spend too much time in a small cage, but your parents know the money & effort for the better cage will not be wasted.