We are in a situation where we need to tile over wood parkay flooring.
"Need to" or just simply want to so as to avoid a little extra labor? Parkay should never be tiled over.
The existing flooring has been glued down and has been there for a very long time.
Has nothing to do with anything.

It can not be removed because it tears up the subfloor and we can't put down 1/4" cement board because of a height issue.
What are the height issues, exactly?
So what can we use to glue the tile down without having it come off and grout cracking?
No way that tile floor should be installed the way you propose to do it.
