I did a quick search of the Social Security Death Index and found the followng:
Richard H. Allen b. 20 Feb 1943 d. 17 Mar 2005last residence Sandy, Salt Lake Co, UT card issued in Arizona
I also did a quick search of yahoo.com"people search"
There were 2 in Salt Lake City, UT as follows: Rich Allen and Rick Allen
plain Richard Allen - in UTAH - there were 9
I would suggest that you not pay the $27.00 fee for a copy of the application for Social Security for Richard H. Allen just yet.
I would suggest that you look into the library at Salt Lake City for Salt Lake County - they usually have obituaries listed in books (our library does). Also you could contact the county for death records. As family, you would be privileged to receive (maybe $5?) a copy of the death certificate.That way you could know if this were your father or not.An obituary in the newspaper would also tell you facts which might help. If you can find out the name of the funeral home, that is even better - they can tell you what family member took care of the arrangements (usually) and that may give you a clue - perhaps it was his brother.
Do you know if he re-married?Perhaps the wife took care of arrangments.
Another suggestion - see if someone can look up the 2000 census for Salt Lake County, Utah - he may be listed there and other information is given also. Another census to check would be the 1970 US Census.
Do you know where your mother and father were married? The marriage license may give clues - if you have a copy. Where were they living when you were born?Did your father attend a Church ?Sometimes Church records hold clues.
I hope for your sake that the Richard H. Allen in the SSDI is not your father - I hope you find him alive and can make contact.
Richard and Allen are not unusual names, so it may be hard to trace him.
Have you ever spoken with his brother? Do you know where his brother lives or lived? That may be a clue also.