It's a hard question to ask, because there are so many variables. Liberalism focuses more on self expression and inclusion, while conservatives focus more on traditions and values.
Media has a lot to do with it, but I'd be more inclined to say that conservative parents have a higher percentage of liberal children than the other way around. I have no research, but coming from the bible belt, I know how constrained and one sided a conservative family can be. Liberal families can have that same mindset, of course, but they're usually more open with their children and allow them to find themselves in a more freeing way, if that makes sense.
Basically, you're shamed if you don't believe what conservatives believe. Liberals, on the other hand, encourage self expression (to a certain degree)