how to connect pre- and post-surveys and link parent & child surveys anonymously at the same time

Are we collecting any unique data on the survey which would help to identify the parent/student individually as well as establish the relation between them?
If yes, then we can use this unique identifier on dashboards and compare data for all four surveys.
If no, then this seems to be not feasible.

We are hoping to be able to provide complete anonymity if possible. So currently there are no unique identifiers.
Another post suggest adding a "Family ID" to connect parent-child dyads. Can that be done to connect the parent-child AND connect each individuals pre- & post-survey at the same time?
If so, what is a "Family ID" and how do we set it up? ... How would be set both up at the same time?

Yes, some unique value is required. Below is a way to achieve this:
The first survey that you give should generate this Family ID using the random number embedded data. At the end of the survey, you can display this ID to the parent/student and ask them to retain it for future surveys. This ID should also be stored in a mailing list as ExternalDataReference using a contact list trigger/XM directory task.
Now for all subsequent surveys, use an authenticator where the respondent needs to enter this Family ID to start the survey. The authenticator will validate the parent/student if a correct ID has been entered.
Finally, this ID can be used to relate responses either within Qualtrics or outside Qualtrics.

How do I make sure they each get the same randomized number ... they are each individually doing different surveys, that need to be connected?

The random number should be generated in the first survey only and retained by the parent and shared with the student. All subsequent surveys should ask the respondent(either parent/student) to enter this random number. When entered, an authenticator will authenticate the user from the contact list which will house these random numbers.