There is no real way to prove they are online with parental permission. We know from the terms of service, a user of 13 MUST have permission to even install the program. I would suggest that we ask, and if they say yes, we trust that. because we can not prove it either way, we have only their word and the fact that its installed to know.
However monitor the behavior. 13 year olds act different in front of mom and dad