I accidentally parked without purchasing a parking permit in a lot at
the University of California, Berkeley campus, and as a result
received a parking ticket.  I've received one parking ticket before in
the past on private property issued by privately employed police and
was never penalized for my failure to pay the ticket.

Since this ticket was issued by the campus police, not by the actual
Berkeley City Police, what kinds of consequences, if any, can result
if I choose not to pay the ticket?

If you are a register student they can put a hold on your records
until it is paid at least that was the way it was done 40 years ago
when I was a student.  The campus police may be a arm of the State
Police and might be able to restrict your ability to register you
vehicle the next time it is do.

I don't think that could happen, since although I am a student, the
car I was driving is not registered under my name but under my
parents'.  There shouldn't be any way they could trace that back to
me.

However, I am wondering whether the UCPD is supported by the State
Police, and thus, whether they actually have that power.  (They claim
to on the ticket, but they could be bluffing...)

You screwed up.  Pay the ticket and get on with your life.