How do I search for public information on the parents of and on the
young man my daughter is dating?

Visit the local county courthouse to do name searches to see if there
are any criminal court cases involving them.  Hire a private
investigator to do a background check for about $100 per person.  Do a
GOOGLE search engine using the person's name in quotation marks and
also check to see if the city they live in has an online newspaper
where you could do a name search through the database to see if there
have been any newspaper stories about them in the past.

Private Investigators have subscriptions to databases that are only
available to licensed professionals. Many Internet sites may provide
clues, but you should really consider a professional. Some of the work
(like visiting a local court house) can be done by anyone and some are
even online. The databases PIs use shave a temendous amount of time
off in most cases because they can do it remotely and cover nationwide
data. I am one of the investigators at RedTeamInvestigations.com.
Average PI fees are $60 an hour, which may be worth it considering you
are looking out for your daughter's interests. Once you get the
details you are looking for, then you may consider hiring someone in
your local area that could investigate anything that would require
surveillance, interviews, etc. Good Luck!