The research on this is pretty muddy, but the answer would seem to be "Some, but not very."
Historically, researchers observed a strong correlation between parent political identity and child political identity - including political identities of "unengaged" and the like. More recent research has observed that a lot of the studies that measured that concordance assumed that the political identity began first with the parent and was passed to the child.
As early as 1968, Jennings and Niemi called out these flaws and suggested alternatives to measuring the directionality of political identity transmission. So while a large congruence was observable, there seemed to be a whole mess of factors that contributed to it, not just simple parent->child transmission.
The factors that contribute to successful parent->child transmission, however, are a little better understood. Jennings, Stoker, and Bowers (2009) did a three-generation study and found that the strength of the political identity, and how active the parents were in politics, contributes greatly to that political identity being acquired by the children.
However, more recent research by Ojeda and Hatemi (2015) finds things like child->parent transmission which - despite the durability of the direct parent->child transmission model, is easily discovered to occur. Any model of the transmissibility of political identity and parent/child relations needs to account for this, and so they open a line of inquiry which they revisited in 2020.
Their model basically says that children will acquire their parents' political identity more frequently not only when that ID is strongly held and regularly performed, but also when the parent->child relationship is supportive and nuturing.
Interestingly, the strength of education seems to induce a sort of rebellion effect where the child will actively reject what they believe their parents' political identity to be. (The child's understanding of their parents' political identity is not necessarily good.)
tl;dr - There is some correlation between parental political identity and that of their children, but the forces that contribute to it are many and strong - parental political ID, even where there is great congruence, is probably not the dominant force causing that congruence, except in healthy, closely held family ties.