I just opened my daycare, so I am new to running my own business, but I definitely believe that the whole point of having a contract is to avoid having clients walk all over you. I would agree that contracts are not negotiable. I know writing up my contract took a lot of work and thought b/c I wanted to be fair both to myself and to parents. If you were to change your contract every time someone didn't see eye-to-eye with you on a policy you would end up with one that reflected the needs of everyone EXCEPT you!
I also think this mother is being unfair. In my contract I stipulate that all stat holidays are paid holidays and if they fall on a weekend I take the Friday or Monday. And while it is true that when a child is sick the mother may have to take a 'financial' hit, you are running a business so why should you have that financial burden instead? Finally, yes you lost a client, but do you really want to work with someone who possibly believes that their time is more valuable than yours?