Hi Dave,
I've just tried it out on 2.5, and had very similar results to my earlier post. It looks as though you may not be unticking all of the submission options - make sure you untick the circled boxes (and any others, if you have additional submission plugins installed) on the assignment settings:
It should then appear like this to students:
Hopefully the information there is easy enough for your students and their parents to understand; if the parents are seeing the "grading summary" information, you should check which role they have been assigned and make sure that they are not a teacher (or non-editing teacher) in the course - otherwise they'd be able to grade their own children's work, which you probably don't want! If they have a role other than those, check that role's permissions to ensure that they don't have permission to grade.
On the due date, after the time that it's due, the upcoming events block looks like this:
and as of the following day, it'll look like this (other assignments notwithstanding):