Let pod_-k(n) denote the number of partition k-tuples of n wherein odd parts are distinct (and even parts are unrestricted). We establish some interesting infinite families of congruences and internal congruences modulo 4, 16, and 5 for pod_-2(n), pod_-4(n), and pod_-6(n), respectively. We also find Ramanujan-type congruences modulo 5 for pod_-3(n) and densities of pod_-2(n), pod_-3(n), pod_-4(n), and pod_-6(n) modulo 4, 5, 16, and 5, respectively.