This might seem like a kind of "secondary" theme, considering that it only pops up in the second part of the poem, and it's almost entirely fictional (we know next to nothing about Trujillo's mother, and Dove admits that she only loosely based the poem on actual history). But in terms of the poem, it's both important and fascinating. The implication in the second part of the poem is that the genocide Trujillo commits is somehow related to the way in which he grieves over his mother's death. As if he is so depressed and angry over her passing that the only way to feel better is to kill other people. This is completely twisted, but the notion of family and loss become integral to the poem, both as a way to understand Dove's imagining of Trujillo's character and as a possible cause for the atrocities he then commits.
The death of Trujillo's mother, and his reaction to it, make the dictator seem more complex and human.
The familial theme present in "Parsley" showcases the dictator's insecurities with his own racial heritage, his ability to retain power, and his lack of control over his own emotions.