How can I cheaply refresh parquet in an apartment?
It really depends how clean the floor is.  Particularly where the finish (laquer) has worn off.  The purpose of sanding off the finish, as well as a thin layer of wood is to produce a clean, new floor look.  

However, it seems that you are trying to just make it livable, not brand new. Here's what I'd do:


Mop the floor well, with some bleach in the water.  This will clean up some of the wood.
Let the floor dry completely. 
Sand with whatever tools you have.  It would be easiest with one of these: Drywall pole sander  if you don't have any power tools.  
Smooth out any obvious bumps, but simply just try to rough up the existing finish so the new finish will adhere.  
Refinish.