-Why wouldn't a nurturant parent want someone else's kid to be fighting in Iraq, so we don't have to fight here?

-Why would a strict father encourage his son to recklessly destroy natural resources at the expense of future generations?

-Why would a nurturant parent want to encourage their child to choose an alternative lifestyle that could cause them to suffer degradation and discrimination for much of their adult life?

-Why must a strict father take serious issue with which gender his child is attracted to?

Unrelated to those three issues - why would a strict father oppose his child getting a decent science education, and why oppose a decent biology education in particular? Why wouldn't a nurturant parent? Why would a nurturant parent want to prevent his (adult) child from carrying a sidearm to protect himself? Why wouldn't a strict father want to prevent this?

When we hear "strict father" and "nurturant parent" those terms make intuitive sense to us, and we then filter information within that framework. Because the terms are vaguely defined, it's easy for us to interpret new information as fitting within this framework. Because there exists a way of describing nearly any view as belonging to either a nurturant parent or a strict father, this framework has no predictive power and very limited descriptive power.

When we are asked to pair "strict father" and "gay rights," the availability heuristic gives us an image of a man with a military haircut throwing his gay son out of the house, or something roughly analagous. With "nurturant parent" and "gay rights," the availability heuristic gives us an image of someone encouraging their gay son to be himself. I would imagine there are plenty of parents we would call nurturant who strongly discourage their kids from "being gay," and there are probably many fathers we would call strict who have no problem with their sons or daughters being gay. The discrepancy is probably much larger for something like environmentalism or health care, and there probably isn't much of a pattern for agricultural subsidies.

There is nothing about strictness or nurturance that dictates any particular political attitude. We simply associate these attitudes with people, and we associate those people with political viewpoints. The reality outside of our simplistic, uninformed mental picture is likely a whole lot more complex.