I'm not naturally loud and outgoing but like to be sociable. When my kids were in primary I did mix with the other parents, though it didn't lead to any longstanding close friends. My children are adopted which made me feel different from the rest of the families, and with my eldest, who had obvious behaviour issues, that led to me feeling uncomfortable, especially as I didn't want to make a point of saying she was adopted (but would if it came up naturally).
In her class a huge group of the mums already knew each other, but they were welcoming and some invited me to events and I chatted at different times with most of them. What helped a lot was that there was a play park very close to the school which many people took their kids to after school so it was a natural place to chat. We virtually all worked part time so weren't always in a rush to be off. And it was a one form entry school so quite easy to eventually work out who everyone was. I did click with one mum, but her dd hated mine so sadly that relationship eventually fizzled out.
I also had a friend from work with an older child at the same school so we often went for coffee etc. (In fact I felt she inhibited my chances at making other friends at times!)
The parents of my younger child's class (same school) were a very different set. No big group that knew each other. I never got to know them all but did chat to most - younger dd got on well with other kids, so was invited out a lot more than her sister so there were more opportunities to mix. We did form a group of friends and had pub evenings etc. That class had far more dads doing the drop off and so we mixed as couples, rather than mums. My youngest is now year 11 - we are still in touch with that group, though it's mainly the men that have kept it going. Dd is still friends with one of the kids from the group though they go to different secondaries. But now they make all their own arrangements and travel by bus I no longer see the parents much.
I never saw myself as successful in making friends from my kids primary but this thread has made me reassess that! Think I did quite well in fact.
But it is hard to sustain once the kids leave primary and you no longer bump into them so regularly. Not one mum has become a 'proper' friend - not even any from the pub group (I knew 2 of the dads much better than their partners). We are FB friends only really, plus chat in the street if we happen to bump into each other.
I'd suggest you look for connections however small, rather than breezily introduce yourself. Think of things to ask or make comments on any shared experiences, however trivial. With one parent, who I still know in a casual way as her dd is at the same school as mine, we bonded over tv series. She wasn't very sociable but we lusted together over various actors and she introduced me to new series I hadn't heard of. Nothing to do with the school!!