Both marquetry and parquetry are types of veneering.
One comment about veneered antiques generally: most people think of veneering as something cheap or done to save money over wood solids. And this is true if you're talking about the 1960's paneling in your rumpus room. But antique decorative veneering was actually an art form perfected in the 18th century (by ebenistes--see below) and a characteristic that added tremendous value to a fine piece of furniture. It was not done to cut corners. It was done to add beauty and value to the best pieces of luxury furniture.
Ok, so the basic concept to understanding marquetry and parquetry is that they are both veneers. Veneers are thin sheets of material that are affixed to the surface of the furniture carcass. Typically, these veneers would be in materials such as ivory, mother of pearl, rosewood, satinwood, purpleheart, ebony, and other exotic woods.
What distinguishes marquetry from parquetry is that marquetry is veneers that create a figural pattern (e.g., people, scenery, flowers, etc.). Parquetry is veneers that create geometric (that is, non-figural) patterns. Simple dimple.
Here is wood parquetry:
More parquetry-here in mother-of-pearl: