Sociologists Dalton Conley and Emily Rauscher claim:
Using nationally-representative data from the  General Social Survey, we [Conley and Rauscher] find that female offspring induce more conservative political identification. We hypothesize that this results from the change in reproductive fitness strategy that daughters may evince.
But economists Andrew Oswald and Nattavudh Powdthavee have found the exact opposite:
We [Oswald and Powdthavee] document evidence that having daughters leads people to be more sympathetic to left-wing parties. Giving birth to sons, by contrast, seems to make people more likely to vote for a right-wing party. Our data, which are primarily from Great Britain, are longitudinal. We also report corroborative results for a German panel.
How to resolve these? See here.
Some of the most left-wing Americans I know have daughters, often nothing but daughters.
If not for those pesky girls, we'd have had a revolution by now!
I'm a 'merkin and have become more liberal as I've gotten older and produced three daughters. I don't know whether it's age, offspring gender, or something else that's made me identify more with the left. Not that I was ever particularly conservative.
This may have a lot to do with points of reference. I have daughters and am, by by native country's (U.S.A.) standards, left-leaning. I would, however, likely be considered moderately conservative by most Europeans. When asked, I call myself a radical centrist.
I, too, am the middle-aged eldest of a four daughter brood, and in no way consider myself right-leaning. And my parents are certainly not right-wingers. How did this happen in Deep South, USA?