I've found that some parents are willing to be convinced about a particular animal and others just won't budge regardless of all the convincing you try to do and if that is the case then you'll just have to accept their rules. There'll be landlords that will be the same and quite often there's no budging room with them either.
Are there any animal shelters that you could volunteer at or does someone who lives in your area have a cat that you could volunteer to look after when they go away or something? It could consider to help convince your parents that you're responsible enough for one and that you're learning as much as you can about the care that they require.