No for many reasons. First, wood, especially parquet style are not suitable subflooring. Slate requires an extremely strong substraight. If you are not on a concrete slab, you must create one with either a 1/2" cement board or a mudbed from metal lathe, sand, and portland cement. Even on concrete with parquet on the top would not work. The moisture that naturally comes through the cement would force the wood up in time as the water has no place to escape to.
Also, the thinsets used on slate tiles would cause the wood to swell. These are mixed with water and can damage a wood subfloor.