parents want to love you no matter what, they want to support you no matter what, they really do. But when they are faced with something that just doesn't fit with what they feel is right... it's very confusing to them. but support for them might end up being the opposite of what you need because of that. And each parent does it in a different way, maybe disowning you, maybe forcing you to be what you were born as...
my parents also say things like "we'll always be there for you", and i'm out to them, but they don't understand they've never been there for me and they still aren't. They say they love me, but they don't know how to show it. But i'm lucky, they try not to hinder my transition and i'm still part of the family.
as for your father, make sure if he really means what he says about LGBT people, sometimes people just think it's fun and there's no harm done, but when it comes down to it haven't really formed an opinion on gay people and so on.