The "they don't really care" argument is funny because really, a party isn't a person with a heart and mind. It doesn't care about anything.
Individuals within parties can care and there are MPs in both the LP and CP who really have strong convictions, including about this issue. Is anyone really saying the MP who wrote this doesn't care? Because she ,and others who have spoken out publicly or in person to people are real people who care about issues.
The parties in some sense encompass these individual feelings but also are something else. There is something like a drive to create a kind of vision or put a stamp on society. For Conservatives that seems more pragmatic, partly at least because that kind of pragmatism about how human society works is built into their ideology. For the LP at the moment it's a sort of pure utopian vision, which is divorced from how real people behave, and how the world is. So - they are disadvantaged when it comes to being elected.
But caring or not caring isn't something an entity like a party can do.