We are starting to remodel my in-laws kitchen and dining area. They
have chosen a self stick vinyl floor tile they like for the flooring.
I will install whatever they want to use, but my gut feeling tells me
those tiles will be problematic.
I have no experience with self stick vinyl floor tiles, but all those
edges concern me. I would think if you were sitting in a chair and
scooted it forward or backward that it would catch the edges of the
tiles and cause damage?

You can add my gut feeling to yours. The industrial one-square-foot
tiles that are not self-stick are fine, given adequate substrate
preparation, but I'd not use the consumer grade stuff. Did that once
in a motel laundry room, at the behest of the manager, but lifespan
and beauty weren't a concern.
Tell 'em to go with sheet vinyl.

Unfortunately, I have not been able to find anyone who makes a sheet
vinyl version of a wood parquet floor.

My mother-in-law found a wood parquet vinyl tile for 88 cents sq/ft at
one of the box stores. It has been hard to get her to look past the price
and point out you "get what you pay for". The kitchen will only be about
150 sq/ft when we're done, so the overall price difference won't be that
significant even with more expensive flooring options.

My wife and I both think the parquet pattern is old and outdated, but her
parents have different tastes. They're both in their 70's with poor
health. They've lived in the run down house for 50+ years, so we're
trying to give them "their" idea of a dream kitchen for the years they
have left. At the same time, it's a lot of work and I would prefer not to
redo it in a few years.

They don't like the look of tile or natural stone at all, and they
haven't really cared for any of the sheet vinyl flooring they have seen.
They are borderline accepting of T&G hardwood floors, but it's hard to
get them to look at anything but those parquet vinyl floor tiles.

So, I was hoping someone could give me something more than my gut feeling
one way or the other how those self stick vinyl tiles would perform.

Thanks,

Anthony