I chatted to a few mothers at pick up time, but those chats never went beyond that. Somehow we never progressed to coffees and real friendships. I worked part-time, so I wasn't at the school every day. Maybe that would have made a difference?
I got to know parents (mothers and fathers) better through birthday parties and afterschool activities. It tended to be the same parents/kids at the same birthday parties and the same dance classes and Scouts etc. And a lot of those parents worked, so I wouldn't have encountered them at the school gates in any case.
The other thing I found is that your child's friendships ebb and flow. DD had a really big friendship group at the end of primary school. But some of those friendships ended when they got to secondary school, even though they moved on to the same secondary school. Some of the girls gravitated towards other kids. There was no big falling out or anything, it just seemed to happen naturally. So I was glad I didn't get too friendly with those mothers - it would have been awkward. Other friendships did survive. And new friends emerged.
There was the odd dispute or falling out over the years (nothing too serious, thankfully) but I never got involved or approached the other parents. I left DD to sort it out for herself. I did give advice though.
The only advice I would give is try not to stress about stuff or get involved in any petty drama - the other parents, the other kids, birthday parties, school politics etc. Obviously serious stuff like bullying needs to be tackled though. Schooldays don't last forever and at some point they'll move on to university or work. DD (now 20) is totally caught up in her adult life - her college course, work, learning to drive, her boyfriend, her plans to travel etc. She still sees her school friends regularly, but hardly thinks about school at all. Those days of waiting at the school gate seem like another life.