He took part in the 1933 British Chess Championship, though he had to drop out through illness after two rounds (he had won the first game and drawn the second).
Parr played in 25 British Chess Championships from 1936 to 1991, compiling an overall score of 134.5/275.
He played several times in the top section of the British Chess Championship in the 1930s (for which only an elite group of twelve players qualified).
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Brian Eley belonged to the wave of talented chess masters who came to the fore in Britain in the 1970s, after the dominance of Jonathan Penrose ended, a group that included Raymond Keene, Bill Hartston, George Botterill, Robert Bellin and others.