Do metastatic cells still have their parent's identity?
Welcome to stack. This is a good question. 

To answer your question: Generally, yes. 

Howeve like @user137 stated below it can depend on many factors most notably how many division cycles they go through, and also where they metastasize to. The immediate environment of a cell is very effective at altering gene expression.

I use Hepg2 cells regularly. These are hepatocellular carcinoma cells that can be cultured. These cells are known to express 3-hydroxy-3-methylglutaryl-CoA reductase and hepatic triglyceride lipase, two enzymes characteristically highly expressed in liver cells. If we want to do a quick quality control typing we will perform rtPCR for those enzymes. I have seen the expression drop off after about 4 months (~35 passages). 

Looking at gene expression (ideally through microarray) is a very valid way to identify/type a cells tissue of origin.