Are my parents homophobic, or it's something else?

They're both fine with me wanting to date and marry a guy in the future. However, I really question their acceptance because of many factors.When I sort of came out to them (I didn't direcly told them I was gay because I am against the idea of labels, but I told them I liked guys), my mom was a little bit shocked not because of my looks or because of how I behaved, but because because I was a quiet kid, and usually people notice when someone's gay when the guy dresses himself as a woman and are loud and obnoxious, according to her. She suspected that I wasn't straight because I didn't date anyone in high school, but she thought I was simply uninterested and too busy with school to date.My dad got silent and didn't speak about my coming out for a long time. He bitched about a pride flag I bought and put on my desk in my bedroom, thinking that it wasn't appropriate, but I still left it there, until last month, when I burned it because I had an immense desire to do it and hide myself again.My mom doesn't like like flamboyant lgbt people at all, and is disgusted to drag queens and everyone who shows any signs of being lgbt, such as people with rainbow flags on their social media profile or clothes etc. She's obsessed with being discreet and she wants me to be the same way.I barely know what my dad thinks of other people, but he seems to be more chill about others, but not so much with his own son.