Is there anything I can do to stop parking on my property or am I going to have to resort to having the cars towed away? Can I even do that?I live...

You pay rent, right?  Depending on your jurisdiction, part
of the rent means that YOU PAY FOR spaces to park (are these numbered parking spaces?). 
Those spaces are likely part of your lease, which gives you equitable title to them -
especially if they are numbered.  Look at your lease.  More than
likely, it is stipulated in your lease that you are also paying for parking spots.  If
that is the case, and the parking spots are designated as yours by number, then most
jurisdictions require that you "give notice" in order to tow at the owner's expense.  So
here's what I would do (you've probably done more than enough,
already):

First, I would go to your landlord with a copy of
your lease and demand that he post signs (if not already posted) and enforce towing. 
I'd say that if he doesn't do it, that you are going to post a sign in front of YOUR
SPACES and then call a towing agency yourself.  You have to post the phone number of
where they can pick it up.

Now, if your spaces are not
numbered, this is a bit more tricky.  You still have a right to the parking - but you
can't exactly take matters into your own hands with the towing.  In this case, you will
have to talk your landlord.  Clearly, this is a management problem with YOUR apartment
(if they were enforcing towing the dance students would know it and would stop parking
there - in college, everyone knows where they can and cannot get away with parking).  If
your landlord is dragging his feet and this problem is a dealbreaker (for me it
definitely would be) be prepared to say you want out of your lease - there is likely a
breach of contract when you have no place to park.  I'd also be ready to say you would
make this problem known to anyone who was looking to move into this building.  As a
student, you are his best advertisement.  Be willing to work with him - but be willing
to stand your ground and demand that things get taken care
of.

I hate this situation for you.  If you know a lawyer,
you might also contact him or her... sometimes it helps to have a legal partner if for
no other reason than to show people that you mean
business.

My final tip - if you want to just take the back
seat on all of this and avoid confrontation - is go buy two of those large orange cones
and put them in the front of your spots.  Sounds silly - but even though people ignore
signs, they often avoid parking where they see cones.  Who knows?  Personally, I'd put
the pressure hard on your landlord to do his job.