Why not volunteer at your local animal shelter? You'll get your fix of cats and learn to care for them at the same time, and your folks will see that you're being responsible - and for yourself, you'll find out if you really do want to have to look after one, without the obligation of … Read more

Lying on their backs shows their vulnerable underparts, I have read that they only do this when they feel happy, safe and secure, so it sounds like you are taking really good care of them and they trust you totally!