There has never been a "burgandy" colored beret associated with a U.S. Military service. Closest would be a Maroon one worn by Paratroopers/Airborne. From a webpage about the History of the U.S. Militay Beret: "Troops of the 82nd Airborne Division at Fort Bragg, N.C., started wearing the maroon beret in 1973, while at Fort Campbell, Ky., the trend exploded, with post personnel wearing red, military police donning light green, and the 101st Airborne Division taking light blue as their color. In Alaska, the 172nd Infantry Brigade began using an olive green beret...Over the next few years, the whole thing got out of hand, and in 1979 senior Army officials put on the brakes, Bradford said. The leadership allowed the Rangers to keep their black berets and in 1980, agreed to allow airborne troops to continue wearing the maroon version. But all others varieties were declared off-limits." http://usmilitary.about.com/cs/generalinfo/a/berethistory.htmhttp://usmilitary.about.com/cs/generalinfo/a/berethistory.htm
So her father must have enlisted or drafted 1973 or after stationed at Fort Bragg, NC as part of 82nd Airborne. (or enlisted and went through boot camp and then at Fort Bragg after adopted berets) Have her register with reunion websites sites of the 82nd companies or units.