Scared of my parents - advice please

So I recently posted a thread somewhere about how I came out to myself as bisexual. It felt amazing, but just a few hours later my parents killed it. Some background - I'm thirteen and they think I'm straight. They were noticing me liking some LGBT related posts on Instagram and some kids from my school who are gay. They said something along the lines of, "We are completely for gay rights, but at the age of thirteen you should distance yourself from all that because there is a sexual component and you shouldn't worry about that."
Funny, when kids say they identify as straight it's no problem and it's not inherently sexual. I was planning to come out to them sometime soon because I don't want to hide, but this changed my mind. I'm terrified they'll say it's a phase or that I shouldn't think about that or even worse blame it on my LGBT friends. I feel like they'll never accept me at this age but I just can't wait to tell someone. Advice?