Parity bias for the partitions with bounded number of appearances of the part of size 1

While there are more partitions of n having more odd parts than even parts compared to partitions having more even parts than odd parts, we show that if there is a limit on the number of appearances of the part of size 1, there will eventually be more partitions of n with more even parts than odd parts compared to partitions with more odd parts than even parts. To demonstrate this, we obtain the asymptotic formulas for related partition functions via a Tauberian theorem, q-series manipulations, and partition combinatorics.