We bought it at an auction about ten years ago. The auctioneer says it is over 100 years old and Belgian. I don't know if that is correct or just sales talk.

The top is parquet, and has two solid extensions that pull out. It looks like it was refinished prior to us having it.

Since we have had it, a glass sweated and left a ring, a hot casserole dish on a trivet left white marks where the heat radiated through the trivet, and now I have to use a vinyl liner under a tablecoth when I use the table, and anything hot has to be left in another room. I have used furniture polish and mayonaise on the white marks and the mayo works for a while but then the white marks come back.

Many years ago, we bought unfinished furniture and used an oil finish on it. It was rubbed on and we put about 7 coats on the table. Water didn't penetrate and hot things on trivets didn't bother it. I looked for that finish several years ago and it is no longer made, but I'm sure there is something similar out there. That table isn't parquet though and that could make a difference.

I bought something at Rockler to remove the white rings, but I haven't tried it yet. If it works, I'd like to add more coats of good finish to try to make this table resistant to white marks. If it doesn't work, I may have to strip the top and start over, but I really don't want to do that. My goal is to be able to use this table like a normal table and not worry about what people are setting on it.

Any suggestions?

Either shellac or lacquer.

Lacquer is usually a little more water resistant but can blush if it is not thick enough on the surface.

Just about any varnish is reasonable water resistant.

The underside of the table should have the same finish.

Use a cotton swab and some denatured alcohol on the underside to see if any of the finish comes off on the swab.
If it is shellac some will come off and the spot will be sticky till the alcohol evaporates.

If nothing happens, it is probably lacquer.

You can do the same test with lacquer thinner.