OK here's how I'd go about it (just a suggestion). On your one day a week, make sure you are early to drop off and pick up. Each time, set yourself the task of chatting to someone, try lots of different people.
Keep it light... 'how's [other parent's child's name] finding year 2'? 'what are you doing for Christmas'? 'man it's been raining so much!' etc.
See if any of those chats run more smoothly, or if you just find yourself liking some parents more than others. You don't want just any friend, but to spend time with someone you like.
There's a woman I see at swimming every week, and I just find myself liking the way she talks with her kids, and the things she says, and her smile. Her vibe. If she was a school mum, I'd try and make friends with her. Other mums - I'm just not as interested.
Over time circle back to the things they've said 'how was it having your mum over Xmas?' or 'is [other parent's child's name] still finding maths hard?
And remember people appreciate honesty and vulnerability eg 'I find it hard when I don't pick up and drop off to know what's going on at school', 'I love x about parenting and find y hard'. Send a few messages about parenting/school things.
Then ask if they want a play date with the kids at the park (hopefully kids like each other!) or a coffee. If they say no, perservere and try with someone else. Not everyone will like you as much as you like them but that doesn't mean you're not fantastic and there will be some people you really get along with out there.
Good luck! It's hard work to make friends as an adult but it's not impossible.