  In his 1984 Memoir of the American Mathematical Society, George Andrews
defined two families of functions, $\phi_k(n)$ and $c\phi_k(n),$ which
enumerate two types of combinatorial objects which Andrews called generalized
Frobenius partitions. As part of that Memoir, Andrews proved a number of
Ramanujan--like congruences satisfied by specific functions within these two
families. In the years that followed, numerous other authors proved similar
results for these functions, often with a view towards a specific choice of the
parameter $k.$ In this brief note, our goal is to identify an {\bf infinite}
family of values of $k$ such that $\phi_k(n)$ is even for all $n$ in a specific
arithmetic progression; in particular, our primary goal in this work is to
prove that, for all positive integers $\ell,$ all primes $p\geq 5,$ and all
values $r,$ $0 < r < p,$ such that $24r+1$ is a quadratic nonresidue modulo
$p,$
  $$
  \phi_{p\ell-1}(pn+r) \equiv 0 \pmod{2}
  $$ for all $n\geq 0.$
  Our proof of this result is truly elementary, relying on a lemma from
Andrews' Memoir, classical $q$--series results, and elementary generating
function manipulations. Such a result, which holds for infinitely many values
of $k,$ is rare in the study of arithmetic properties satisfied by generalized
Frobenius partitions, primarily because of the unwieldy nature of the
generating functions in question.
