Parity Results for p-Regular Partitions with Distinct Parts

We consider the partition function b'_p(n), which counts the number of partitions of the integer n into distinct parts with no part divisible by the prime p. We prove the following: Let p be a prime greater than 3 and let r be an integer between 1 and p - 1, inclusively, such that 24r + 1 is a quadratic nonresidue modulo p. Then, for all nonnegative integers n, b'_p(pn + r) ≡ 0 (mod 2).