I've only used it on a test basis, so you may get a more expert reply, but in answer to your Qs:
1. Yes -but see 2 below.
2. Yes - assuming you set it up as in Yu' s post in http://moodle.org/mod/forum
But you can alter the role settings for 'parent' to allow/disallow what they can see or do.
3. This happens automatically if you link the parent to the student as recommended above.
4. Yes - if you view 'Participants' in the course you can view 'all' or for each separate role - so you can see a list of student participants and a list of parent participants
Setting up the parent role takes a little fiddling with the role settings, but is not difficult - the list of settings is intimidating but those blue Q marks give you the help you should need.
The fiddly bit is that (obviously) you need to allocate each parent to their offspring.
Hope this helps.