There is a difference between parties that have traditionally advocated for those issues of rights and environment (Liberal, NDP) and those who are dragged along by opinion polls and modern civilization (Conservative). 
The fact remains that voters who are uncomfortable with women's rights activists and the LGBT community, and claim that climate change is a scam, find a place in conservative parties not liberal ones. Such is true beyond our borders as well. 
The shoe fits though you now pretend otherwise. I still fondly remember when Ontario Conservative leader Patrick Brown announced his belief in the science of climate change, threads here were full of wailing from the right: Oh no, who do we vote for now?!