I understand why parents would say "it's just a phase". While (I hope) most parents love their childeren regardless of sexuality, being straight is just "easier". It's (I believe) never an issue being straight, while being gay is just simply not completely accepted by society as a whole. I would love my childeren, but I wouldn't want them to go through of the feeling of not being accepted. If being gay was just accepted by everyone, would I mind? Of course not. You can't expect your child to never get hurt, but you don't want them to either.
For transsexuality: little kids can have a period where they refuse everything associated with their sex. I was a really boyish kid, played with cars and hated dolls, never wanted to wear skirts and preferred boys clothing. I identified more with boys but I grew out to be quite feminine. If someone told my mother that I might be transsexual, she would have probably said that it was just a phase. While this is of course different for transsexuals, I can understand why a parent would think it's a phase and also: the process of changing gender is not easy and you wouldn't want your child to go through such a heavy life experience.
My last point: Sexuality is a lot more complicated for women than for men. Brains from homosexual men and straight men are different and it can be proven where homosexuality comes from (while in women, these things have never been proven to my knowledge). For men sexuality tends to be easier: Men think more in "boxes", they like men, women or they like both. Women are just more complicated in that view: some women prefer men mostly, but have fallen in love with a woman a few times, like sex with a boyfriend but not sex with a girlfriend, etc. Perhaps this is why female friends are often more familiar and intimate with each other than males: While I don't have feelings for my friend (Though I'm quite sure I'm bisexual), I hug her, hold hands with her when I walk around, and we cuddle when watching tv.
Sexuality, gender, sex is just so complicated. It's hard to not offend anyone, while sometimes people read insults everywhere and you can't even get your point across anymore.