How to cover parquet floor with vinyl tiles?

the answer depends on how long you want the floor to last. are you just fixing up to sell or to rent out? If its short term just put down the plywood and go over. It will last for a year or maybe 2. if its your home and you plan on living there for a few years you need to take up the old wood and put the tile on the concrete. The wood will rot if trapped between wet concrete and a non breathing vinyl layer.

If the parquet is installed on a slab to cover it with luan is going to be a problem. You'll have a problem getting nails or staples to hold. Maybe you can fill the areas in the parquet that are separating and the install the vinyl tile. Other wise just tear up the old parquet, and this will be a job especially removing the old glue residue.

I would feather float the floor with armstrong feather float, this is what I do for sheet vinyl and for tiles, also use a good durabond adhesive with the tiles, but do not put too much because it will seep through the cracks,use the smallest trowel you can find and spread it on, let stand a few minutes and place the tile, this has always worked for me.

I would either rip up the floor and lay tile on the slab and if that isn't an opt. then yes I would lay 1/4" Luan over it and GLUE it down and then go that route.

if your parquet is true parquet flooring ie. wood --why would you cover it? shine it up & enjoy!!!!!