First of all, you are not pathetic or weak, let's just get that straight!
I think sometimes the problem with social anxieties is that we feel anxious, but we also feel like everyone else is totally fine and not anxious at all. I'm over 30, have pretty good self-confidence, have no problem speaking to people when I order a coffee or interact with strangers, but I get super stressed when I'm meant to just 'be me' and make new friends. From the outside though, you'd think I was a natural, and that I had this figured out a long time ago; unfortunately not in my head! So while it seems like everyone else is just nailing it (as we often feel that's the vibe people put out, somewhat skillfully!), there are so many people like you and me who just aren't quite sure how to behave when we're nervous. So try not to ignore that idea, that in fact normal people aren't fine. Some are fine, but plenty are just good at LOOKING fine, while they're thinking the exact same things as you and me.
Now, I don't mean to say that your parents are wrong, but I don't believe you need to put in effort to make friends. Yes you need to turn up, and once you make friends, maintaining those friendships can take effort, but just feeling a connection with nice people around you isn't a question of effort. If you don't feel you fit in, or if they're not your people, and you don't feel you can be yourself, then trying harder isn't going to fix that; you've already tried hard by turning up! You know what it feels like to have friends, because you have the one you go to study group with, and I'm sure that didn't take a whole lot of effort to connect with her.
I'm not sure how old you are, but if you mention study group I'm thinking high school or university, and during that time, it can be hard to know how you want to behave anyway. I certainly behaved in high school in ways that now I consider "not me", but we are all just figuring it out along the way, so try not to give yourself a hard time about who you want to present yourself as to others, and just try to take a deep breath, and enjoy being with your friend, and when you are feeling more relaxed and less anxious, the friends will naturally find you :)
Hang in there,