Parma was born in Ljubljana, Slovenia (then Yugoslavia). He first played in the in 1959, sharing second place. Two years later at age 21 he won the next Junior Championship (The Hague 1961), receiving the title of .
FIDE granted him the grandmaster title based on his outstanding performance at the Beverwijk tournament in 1963. He was the third Slovene to become a grandmaster, after (1950) and (1953). He won the in 1959 and 1961 and shared third place with , , and in the 1968 in .