Whether the tiles have to come up depends on a lot of things. Personally, I nearly always advocate taking existing floor coverings up, checking sub-floor for moisture content and starting from there. As for asbestos content, I don’t know, I’ve been on a few asbestos awareness courses now, and if it’s involved, I’d walk away from that part of the job. I just don’t know enough about the procedures for uplift and disposal.
Finally, a parquet floor is one of the best floors you can have and if done properly, will last as long as the house is standing. However, and this is with the greatest of respect, it’s one wood floor that isn’t really for the DIYer/amateur. I’m not saying, “No, you can’t…” but they are hard work and if **** up, a bugger to put right.