Hi Helen and welcome. I would not wash your unquilted front and back pieces. You will end up with a mess of tangles and strings.
As to washing the batting...with 100% cotton batting you will get some shrinkage. If that is not the look you prefer - and many don't - depending on what the package directions say, you can wash/pre-shrink the batting. You have to be very careful thought that you don't end up with a mess of wet cotton in your washer! Read the directions on your packaging carefully before attempting.
There was a thread here on the board about this just yesterday. I'm horrible at being able to find those things but I'm sure someone will come along shortly and provide a link for you.
Good luck with your quilt and hope your parents will be on the mend soon.