Arithmetic properties of a partition pair function

For a positive integer n, let ped(n) be the number of partitions of n where the even parts are distinct, and po(n) be the number of overpartitions of n into odd parts. Moreover, let Q(n) denote the number of the partition pairs of n into two colors (say, red and blue), where the parts colored red satisfy restrictions of partitions counted by ped(n), while the parts colored blue satisfy restrictions of partitions counted by po(n). We establish several congruences for Q(n). We also obtain an asymptotic formula for Q(n).