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) drop 0 (mod 2).