I had a very similar issue last week.
My UNIX timestamp was parsed but it was either parsed with a grokparsefailure or my Kibana simply didn't show me the field as a Date, only as the literal UNIX Timestamp string.
Oddly enough the pipeline that works for me looks almost exactly like yours, except that I use "UNIX" instead of "UNIX_MS"
Have you already tried it that way?
date { match => [ "applicationDate", "UNIX"] target => ["applicationDate"] }