Modulo 2 congruences for partitions with initial repetitions (Q6080932)

\textit{G. E. Andrews} [Acta Math. Sin., Engl. Ser. 25, No. 9, 1437--1442 (2009; Zbl 1189.05018)] defined a partition with initial \(k\)-repetitions as one in which if \(j\) appears at least \(k\) times, all positive integers less than \(j\) appear at least \(k\) times. In this paper, the authors introduce parity formulas for several functions for this class of partitions and present an infinite family of Ramanujan-type congruences modulo \(2\).