Here's how you can enable Parents to update specific information in their profiles.
First, go to Constituent Manager > Setttings > [role - in this case, "Parent"] > Profile Fields tab.
Find the group of fields labeled "Contact Information," and click the little "e" link on the right.
In the window that pops up, select the checkbox at the top labeled "Enable updates to contact information by group constituents."
At this point, you might also want to select the checkbox underneath labeled "Allow this email to be hidden or shown by group constituents" - this will let parents decide if they want to hide their email from the public directory, while still leaving it in the database so that they can subscribe to mailing lists, etc.
- Scroll to the bottom of the window and click "Update Properties" to finalize the change.
To verify that this worked, log in as a parent and log out of your site admin session(s), and click the "Profile" link in the account bar.
In the window that popped up, you should see the little blue pencil icon next to the email address when viewing the parent's profile info - clicking that pencil icon allows users to edit the email address saved there.
You can use the same basic procedure to give parents the ability to edit other pieces of information in their profile, if you wish.