There probably isn't enough specificity in your inquiry to give you the kind of exact information you're seeking. I count at least 31 ships sunk in the Gulf of Mexico during World War II, and at least another 155 sunk in the Caribbean (see http://usmm.org/shipsunkdamaged.html).
First of all, what is the exact name of your uncle? I'm reading loe (Joe? Joseph?) eugee (Eugene?) Junior Parsons? Was he a Junior or was Junior only a nickname?
Second, was he in the Navy Armed Guard or was he in the U.S. merchant marine?
Your best approach may be to attempt to obtain a copy of your uncle's service record, which should show the ships in which he served. Presumably the record would show a ship that was sunk, and/or whether he was injured or hospitalized. (If not, you can research the names of his ships.) Depending on whether he was in the U.S. Navy Armed Guard or the U.S. merchant marine, there are different repositories for these records. See this page from the Armed Guard website, http://armed-guard.com/searchmil.html, in particular section II, Researching Military History Records. This page has information on where to write to obtain service records.
Additionally, since you are not next of kin to your uncle (next of kin = parent, spouse, sibling, child), you should use someone who is next of kin, if any survive (presumably your father or mother would be next of kin to your uncle), to make any formal request for records. You can do the legwork of drafting letters or preparing forms, but someone who is next of kin should sign. If no next of kin is available, make the request yourself and hope for the best.
Ron Carlson, Webmaster
Armed Guard / Merchant Marine website