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Two comments on this:
 Berry does not specify which particular encoding of the Booleans you should choose, only that one must be assumed beforehand, which is completely fair, as otherwise you would never know what true would be and what false would be. So, higher-type (whatever that would be in the untyped lambda-calculus) is covered by the theorem.
 There is adifference between (a) parallel execution, which exists for the lambda calculus in the form of parallel reductions, sharing graphs, etc., and (b) a true parallel program which of which POR is the prime example: just start evaluating both arguments and when one eventually evaluates reduces to ""true"" we don't need to evaluate the other argument any further before we move on to the next part of our ""program""; Berry's theorem shows that this ""we don't need to evaluate the other argument any further"" is impossible. Unfortunately, the most common reading of ""parallel"" is (b) and, to make things worse, at lot of people assume ""concurrency"" is synonymous with parallelism.  "