I find parents loosen up when I involve their kids in conversation, because kids don’t really care and will talk to anyone(!) – by this I mean, for example: a boy at daycare yesterday was telling me about someone else’s shoes that were based on the Narita Express, so I exclaimed how cool they were, asked him if he likes trains, asked him if he enjoyed going on the Narita Express with his dad, that kind of thing (while obviously looking after my daughter too). Then his mum was really friendly toward me and felt more confident asking me questions – because she’d been able to hear my level of Japanese during my interactions with her son, rather than feeling the old “does she speak Japanese? maybe I shouldn’t say anything…” awkwardness.
However, I feel that daycare is for my kid to make friends, not the grownups – everyone’s at hoikuen because their parents have full-time obligations and have busy lives, so I don’t feel any personal need to be super close with any of the parents or exchange contact details or whatnot. Just a polite good morning or otsukaresama desu at the end of the day is enough, and if they don’t respond that’s on them and their bad manners, and it’s not worth the effort of trying to change their whole personalities! The good friendships are the ones that develop organically, after all.